ELEMENTS 


QA  ^L 
CONSTRUCTIVE  GEOMETRY 


OF 


WILLIAM   NOETLING 


UC-NRLF 


*B    527    fi71 


SlLVEF^BUI^DETT  5c COMPANY 


IN  MEMORIAM 
FLORIAN  CAJORI 


tfy>a^- 


KUvMKNTS 


(  ONSTRUCTIVE  GEOMETRY 


INDUCTIVELY   PRESENTED 


BY 

WILLIAM    BTOETLING,  A.M..  C.E. 
a i  raoa  oi  oir  cm   icmci  and  art  of  education 


F720H  77//-;  GERMAN  OF  K.  H.  STOCKER 


SILVER,  BURDETT  AND  COMPANY 

New  Yohk         H<  >ST0N        Chicago 
1897 


Copyright,  1897, 
By  SILVER,   BUEDETT  &  COMPANY. 


NorfoooU  Press 

J.  S.  Cushing  &  Co.  -  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


T^ 


PREFACE. 


Believing  that  there  is  room  in  our  schools  for 
an  elementary  inductive  work  on  geometry,  I  have 
taken  the  matter  of  one  of  the  best  German  books  and, 
as  much  as  possible,  changed  the  method  from  the 
deductive  to  the  inductive. 

The  method  is  constructive,  the  pupils  drawing  their 
own  figures.  It  begins  with  Bimple  exercises,  which 
increase  in  difficulty  by  almost  imperceptible  steps. 

The  treatment  of  the  subject  will,  I  think,  be  found 
thoroughly  educative  and  practical,  and  an  excellent 
introduction  to  demonstrative  geometry. 

The  introduction  of  this  kind  of  geometry  into  the 
country  schools,  where  geometry  has  heretofore  not 
been  taught,  and  into  the  lower  grades  of  village  and 
city  schools,  will  furnish  a  sort  of  knowledge  to  the 
pupils  of  those  schools  that  will  prove  valuable  to  them 
in  the  various  callings  of  life;  a  knowledge,  too,  of 
which  they  are  now  left  in  total  ignorance. 

The  work  is  intended  for  children  from  ten,  or  even 
nine,  years  of  age  upward. 


M306206 


4  PREFACE. 

It  may  seem  to  some  into  whose  hands  this  book 
falls  that  diagrams  should  have  been  given  here  and 
there  in  the  text ;  but  I  believe  that  intelligent  teach- 
ers would  rather  not  have  them  in  the  book,  and 
therefore  I  have  omitted  them.  The  figures  whose 
construction  may  puzzle  some  teachers  a  little,  can  be 
found  in  almost  all  of  the  large  dictionaries. 

The  instruments  that  are  needed  to  make  the  con- 
structions and  measurements  are  a  foot  rule,  a  pair  of 
compasses,  and  a  semi-circular  protractor. 

WILLIAM  NOETLING. 

State  Normal  School, 

Bloomsburg,  Pa.,  June  1,  1897. 


CONTENTS. 


PAGE 

Preface  3 

I.     Links 7 

Length,  Divisions,  Direction 7 

Kiiuis,  how  named 7,  8 

Relation! 8 

II.      As    i  i  > 9 

How  named  and  measured 9 

Kinds 9,  10,  12 

Divisions  of  Circle 10 

Comparison 11-13 

III.  Intersected  Parallels 13 

Names  of  Angles 13,14 

Comparison  and  Measure  of  Angles    ...         14,  15 
Angles  of  Converging  and  Diverging  Lines         .        .       15 

IV.  Triangles 16 

1.  Their  Angles 16,  17 

K elation  of  Angles  and  Sides       ....       16 
Comparison  of  Angles  .        .        .        .         17,  18 

2.  Their  Sides 19 

Unequal  Sides 19 

Two  Equal  Sides 19 

Three  Bqnal  Sides 20 

Congruence,  or  Agreement,  of  Triangles     .        .21 

V.     Fo[  i:-.ii.ii.  FlOUSM 22 

Definitions 23 

1.  Parallelogram 23 

Angles  of  Parallelograms 23 

Rectangle  and  R h<  »i nl  >o id 23 

Bojoaxe  and  Rhomboa 24 

Sides  and  Diagonal 24 

2.  Symmetrical  Trapezoid 26 

VI       I'm,   Poi  roOH 27 

5 


CONTENTS. 


VII. 


The  Areas  op  Figures  inclosed  by  Straight  Lines 

1.  Rectangle 

Kinds  and  measurements     . 

2.  Square 

3.  Oblique-angled  Parallelogram 

4.  Triangle 

Triangles  of  same  base  and  altitude 
Right-angled  Triangle 
Isosceles  Triangle 

5.  Trapezoid 

6.  Polygon  ...... 


VIII. 


IX. 


The  Circle 

1.  Peripheral  and  Eccentric  Angles 

2.  Diameter  and  Circumference  . 

3.  Area  of  Circle 

4.  Annulus 

5.  Sector      

6.  Segment 


The  Fundamental  Mathematical  Bodies 
Introduction  and  Definitions    . 
The  Surface  of  Geometrical  Bodies  . 

1.  The  Regular  Bodies 

The  Cube     .... 

The  Tetrahedron 

The  Octahedron  . 

The  Icosahedron  . 

The  Dodecahedron 

The  Sphere  .... 

2.  The  Half-regular,  Uniformly-thick 

The  Prism,  or  Pillar    . 

The  Round  Pillar,  or  Cylinder 

3.  The  Half-regular  Tapering  Bodies 

The  Pyramid 

The  Cone     .... 

4.  The  Truncated,  or  Shortened  Bodies 

The  Shortened  Square  Pyramid 
The  Shortened  Cone    . 
The  Contents  of  Solids  or  Bodies 

1.  Bodies  of  Uniform  Thickness 

2.  Tapering  or  Pointed  Bodies     . 

3.  Shortened  Bodies     . 

4.  Regular  Bodies 

5.  Ball,  or  Sphere 


Bodies 


ELEMENTS   OF 
CONSTRUCTIVE   GEOMETRY. 


>>Kc 


I.  LINES. 

1.  Draw  a  free-hand  Line  (i.c  without  ruler)  upon  the  black- 
board or  paper,  making  it,  according  to  your  judgment,  one 
inch  in  Length.    Compare  its  length  with  an  inch  measure. 

Note.  —  All  lines  not  otherwise  designated  are  to  be  considered 

Jit. 

2.  Draw  free-hand  lines  two,  three,  four,  etc.,  inches  in 
Length,  and  compare  their  Lengths  with  the  corresponding 
measures. 

3.  Draw  a  free-hand  Line  <>f  given  length,  and  compare  it 
with  its  corresponding  measure1;  then  bisect  it  and  trised 
it.  and  compare  the  Lengths  of  the  divisions. 

4.  Draw  lines  of  various  lengths,  divide  them  into  two, 
three,  four,  five,  six,  etc.,  equal  parts,  and  compare  the  part 
of  each. 

5.  Suspend  a  small  weight  from  the  end  of  a  cord,  and 
draw  a  line  on  the  blackboard  that,  shall  take  the  same 
direction  as  the  cord.  What  name  is  applied  to  lines  that 
take  tins  direction '.' 

6.  What  name  is  applied  to  a  line  that  takes  the  direc- 
tion of  the  surface  of  standing  water?  What  is  the  differ- 
ence between  horizontal  ami  level  ? 

7 


8  CONSTRUCTIVE    GEOMETRY. 

7.  Draw  a  vertical  line  and  a  horizontal  line  upon  the 
blackboard.  Can  you  draw  a  line  that  is  neither  horizontal 
nor  vertical  ?  Draw  such  a  line.  What  name  is  applied 
to  it? 

8.  If  I  should  mark  a  point  upon  the  blackboard,  and 
ask  you  to  draw  a  line  through  it,  would  you  know  what 
direction  it  should  take  ?  If  I  should  mark  two  points,  and 
tell  you  to  draw  a  line  through  them,  would  you  know  in 
what  direction  to  draw  it  ?  How  many  points  are  necessary 
to  determine  the  direction  of  a  straight  line  ?  Would  more 
points  be  of  any  service  ?     In  either  case,  why  ? 

9.  What  name  would  you  give  to  a  line  made  partly  of 
straight  and  partly  of  curved  parts  ?  What,  to  one  made 
of  straight  parts  taking  different  directions  ? 

10.  If  you  were  required  to  go  to  a  certain  place,  and  had 
the  choice  of  three  roads  —  one  of  them  partly  curved  and 
partly  straight,  another  of  straight  parts  taking  different 
directions,  and  the  third  entirely  straight — which  one  of 
them  could  you  travel  in  the  shortest  time  ?  Why  ?  What 
kind  of  line  is  the  shortest  distance  between  two  points  ? 

11.  How  could  you  determine  the  length  of  a  mixed  line  ? 
How  of  a  broken  line  ? 

Remark.  — Lines  are  designated  by  means  of  letters  placed  at  their 
extremities.  Thus,  a  line  with  the  letter  a  at  one  extremity  and  b  at 
the  other,  is  termed  the  line  ab. 

12.  Draw  two  lines  that  shall  throughout  their  lengths 
be  equally  distant  from  each  other.  What  name  is  applied 
to  such  lines  ?  May  more  than  two  lines  be  parallel  to  one 
another  ? 

Remark. — Diverging  lines  are  such  as  separate  more  and  more 
the  farther  they  are  produced,  and  converging  lines  such  as  approach 
each  other  more  and  more  the  farther  they  are  extended. 


A  SULKS 


II.  ANGLES. 

1.  Dmw  two converging  lines,  and  extend  them  until  they 
meet.     The  opening  between  the  lines  is  called  an  angle) 

the  lines  are  the  sides  of  the  angle,  and  the  point  in  which 
the  lines  meet  is  its  vertex, 

2.  Angles  are  named  by  means  of  letters  placed  at  their 
and  at  the  vertex. 

3.  Place  the  letter  a  at  the  extremity  of  one  of  the  sides 
bi  the  angle  you  have  formed,  b  (outside  of  the  angle)  at 
the  vertex,  and  c  at  the  extremity  of  the  other  side.  Thus, 
the  angle  is  named  the  angle  abc;  the  middle  letter  being  the 
one  at  the  vertex. 

4.  An  angle  may  also  be  named  by  a  letter  placed  be- 
tween its  sides  near  their  intersection. 

5.  With  the  vertex  of  the  angle  you  have  formed,  as  a 
center,  describe  a  circle  whose  circumference  shall  cut  both 
sides  of  the  angle.  The  arc  (part  of  ei re n inference)  of  the 
circle  between  the  sides  is  the  measure  of  the  angle. 

6.  With  any  radius,  describe  a  circle.  Divide  its  circum- 
ference into  fonr  e^nal  parts,  and  from  two  of  the  points,  a 

fourth  of  the  circumference  apart,  draw  lines  to  the  center, 
forming  a  center  angle  whose  measure  is  a  fourth  of  the 
circumference.  An  angle  whose  measure  is  one  fourth  of  a 
circumference  is  a  right  angle,  and  its  sides  are  perpendicular 
to  each  other. 

7.  Draw  a  line,  and  at  its  middle  point  erect  a  perpendic- 
ular to  it.  How  do  the  angles  on  each  side  of  the  perpen- 
dicular compare  with  each  other  in  size'.'  What  kind  of 
angles  are  they? 

8.  What  does  OCUU  mean  '.'      What,  obtuse  9 


10  CONSTRUCTIVE  GEOMETRY. 

9.    A  center  angle  that  is  less  than  a  right  angle  is  an 
acute  angle,  and  one  that  is  greater  is  an  obtuse  angle. 

10.  Make  an  acute  angle ;  also  an  obtuse  angle.  How  do 
the  sides  of  these  angles  meet  each  other  ? 

11.  All  angles  not  right  angles  are  oblique  angles. 

12.  Make  a  center  angle  whose  sides  are  a  half  circum- 
ference apart.  What  kind  of  line  do  the  sides  form  ?  Such 
an  angle  is  called  a  straight  angle. 

13.  An  angle  whose  measure  is  more  than  a  half  circum- 
ference is  a  convex  angle.  A  convex  angle  presents  a  corner 
at  the  vertex ;  while  right,  acute,  and  obtuse  angles  present 
angles. 

14.  Every  circle  or  circumference  is  supposed  to  be 
divided  into  360  equal  parts,  called  degrees.  How  many 
degrees  in  a  right  angle  ?     In  a  straight  angle  ? 

15.  Thirty  degrees  (written  30°)  equal  what  part  of  a 
right  angle  ? 

16.  An  arc  of  45°  measures  what  kind  of  angle  ?  What 
part  of  a  right  angle  ? 

17.  Name  the  angle  measured  by  75°;  also  that  con- 
tained between  two  lines  120°  apart. 

18.  One  fifth  of  the  circumference  measures  what  kind 
of  angle  ?     One  third  measures  what  kind  ? 

19.  What  kind  of  angle  do  the  hour  and  minute  hands  of 
a  clock  make  with  each  other  at  3  o'clock  ?     At  6  ?     At  9  ? 

20.  How  long  does  it  take  the  minute  hand  to  pass  over 
the  arc  of  an  acute  angle  ?  Of  a  right  angle  ?  Of  an  obtuse 
angle  ?     Of  a  straight  angle  ?     Of  a  convex  angle  ? 

21.  If  a  man  who  had  been  going  due  east  should  change 
his  direction  three  right  angles  to  the  left,  in  what  direction 


AVQL1  11 

would  he  then  !»••  going?  If  he  had  been  going  north,  and 
had  changed  his  direction  one  right  angle  to  the  right,  in 
what  direction  would  he  be  going? 

22.  How  many  degrees  in  an  angle  that  is  10°,  IS  ,  25  , 
or  60°  leaa  than  a  right  angle? 

23.  How  many  degrees  in  an  angle  that  is  15°,  24°,  35°, 
•  •I-  7.v  larger  than  a  right  angle? 

24.  A  straight  angle  is  how  many  times  as  large  as  an 
angle  <»f  12°,  18°,  45°,  or  60°  ? 

25.  How  does  a  right  angle  compare  with  an  angle  of 
L'7<>  ?  How  does  a  straight  angle  compare  with  an  angle 
of  -70°  ?  How  does  an  angle  45°  larger  or  smaller  than  a 
right  angle  compare  with  one  of  270°?  One  22£°  larger  or 
smaller  than  a  straight  angle  ? 

26.  Draw  an  angle;  also  the  arc  that  measures  it.  With 
the  Opening  of  the  dividers,  with  which  the  arc  was  drawn, 
describe  part  of  a  circle;  upon  it  lay  off  the  measuring  arc 
of  the  angle,  and  from  its  center  draw  lines  to  its  extremi- 
ties, thus  forming  a  second  angle.  How  do  the  two  angles 
compare  in  size  ? 

Note.  —  All  angles  of  the  same  problem  or  figure  (geometrical) 
should  have  their  measures  described  with  the  same  opening  of  the 

dividers. 

27.  Make  an  angle  of  22£°.     What  part  of  a  right  angle 

is  it  7 

28.  Make  two  angles,  of  which  the  second  shall  be  twice 
as  large  as  the  first. 

29.  Make  an  angle  that  is  as  large  as  the  sum  of  a  half 
and  a  fourth  of  a  right  angle. 

30.  Make  two  unequal  angles,  and  a  third  that  shall  be 
equal  to  their  difference. 


12  CONSTRUCTIVE  GEOMETRY. 

31.  Make  an  obtuse  angle.  The  angle  between  its  sides 
is  a  concave  or  hollow  angle  (an  angle  less  than  a  straight 
angle).  What  is  the  sum  of  the  two  angles,  the  concave 
and  the  convex  ?  How  many  right  angles  in  their  sum  ? 
If  the  concave  angle  is  38°,  62°,  84°,  or  125°,  what  is  the 
convex  angle  ?  If  the  convex  angle  is  195°,  225°,  or  312°, 
what  is  the  concave  angle  ? 

32.  Form  several  angles  around  a  point,  and  find  how 
many  right  angles  their  sum  contains.  How  many  right 
angles  can  be  constructed  around  a  point?  How  many 
degrees  in  all  the  angles  around  a  point  ? 

33.  What  does  adjacent  mean  ? 

34.  Construct  an  angle,  and  prolong  one  of  its  sides  be- 
yond the  vertex,  thus  forming  another  angle  adjacent  to  the 
first.     What  is  the  measure,  or  sum,  of  both  ? 

Note.  —  Two  angles  that  have  a  common  side  and  the  other  two 
sides  forming  a  straight  line  are  adjacent  angles. 

3§.   What  is  the  sum  of  any  two  adjacent  angles  ? 

36.  When  one  of  two  adjacent  angles  is  21°,  75°,  or  68°, 
what  is  the  other  ? 

37.  What  is  the  measure  of  each  of  two  equal  adjacent 
angles  ?     How  does  the  common  side  meet  the  others  ? 

38.  What  is  the  difference  between  perpendicular  and  ver- 
tical ?    Illustrate  it  upon  the  blackboard. 

Remark.  — Two  angles,  the  sum  of  whose  measures  is  one  fourth  of 
a  circumference  (90°),  are  complements  of  each  other;  and  two,  the 
sum  of  whose  measures  is  a  half  circumference  (180°),  are  supplements 
of  each  other. 

39.  Construct  an  angle,  and  prolong  its  sides  beyond  the 
vertex.  The  angle  formed  by  the  prolongation  of  the  lines 
and  the  first  angle  are  called  opposite  angles.     How  do  the 


TNTBB8ECTED   PABALLSL8.  13 

opposite  angles  compare  in  sise?  Compare  other  opposite 
angles.  What  general  inference  may  be  drawn  concerning 
opposite  angles  ? 

liniutK.  —  If  the  pupils  have  a  sufficient  knowledge  of  algebra,  the 
relation  <>f  opposite  and  vertical  angta  may  be  shown  by  means  of  the 

c<iuati<m. 

40.    When  an  angle  adjacent  to  two  opposite  angles  is  92°, 
120°,  112°,  or  1  16°,  what  is  each  of  the  opposite  angles  ? 

III.   INTERSECTED  PARALLELS. 

1.  When  are  two  lines  parallel  to  each  other  ?  Draw 
two  parallel  lines.  Draw  an  oblique  line  across  them.  The 
oblique  line  is  called  a  trantverwl. 

2.  How  many  angles  does  the  transversal  form  with  the 
parallels  ?     Indicate  each  of  the  angles  by  a  letter. 

3.  Name  two  angles  that  lie  within  the  parallels  and  on 
the  same  side  of  the  transversal.  Angles  that  lie  within 
tin*  parallels  arc  called  interior  "n<jles.  Name  the  other  pair 
of  interior  angles. 

4.  Name  two  angles  that  lie  on  the  same  side  of  the 
transversal,  but  without  the  parallels.  Angles  that  lie 
without  the  parallels  are  called  exterior  arujles.  Point  out 
another  pair  of  exterior  angles. 

5.  Find  two  angles  on  the  same  side  of  the  transversal 
—  not  at  the  same  intersection — one  within  the  parallels, 
the  other  without.  Such  angles  are  called  corresponding 
angU*     Name  three  other  pairs  of  corresponding  angles. 

6.  Which  two  interior  angles  lie  on  opposite  sides  of  the 
transversal  and  at  different  intersections  ?  Such  angles  are 
called  aUerncUk  into  rior  angles.  Point  out  another  pair  of 
alternate  interior  angles. 


14  CONSTRUCTIVE  GEOMETRY. 

7.  What  two  exterior  angles  can  you  find  that  are 
neither  on  the  same  side  of  the  transversal  nor  at  the 
same  intersection  ?  Such  angles  are  called  alternate  exte- 
rior angles.     Find  another  pair  of  alternate  exterior  angles. 

8.  Find  two  angles,  one  exterior,  the  other  interior,  on 
different  sides  of  the  transversal  and  at  different  intersec- 
tions. Such  angles  are  called  alternate  opposite  or  conjugate 
angles. 

9.  Find  two  angles  on  the  same  side  of  the  transversal, 
either  both  within  or  without  the  parallels,  and  at  different 
intersections.     Such  angles  are  called  opposite  angles. 

10.  How  do  the  measures  of  the  corresponding  angles 
compare  ?  How,  those  of  the  alternate  exterior  angles  ? 
Of  the  alternate  interior  angles  ?  What  inference  may  be 
drawn  concerning  the  corresponding,  the  alternate  exterior, 
and  the  alternate  interior  angles  of  two  parallels  crossed  by 
a  transversal  ? 

11.  Describe  a  semicircle  (half  circle),  and  from  one  end 
of  its  curve,  with  the  same  opening  of  the  dividers,  lay 
off  an  angle  equal  to  the  sum  of  a  pair  of  interior  angles, 
and  observe  what  part  of  the  semicircumference  it  meas- 
ures. In  the  same  manner  determine  the  measure  of  the 
sum  of  a  pair  of  exterior,  also  of  a  pair  of  conjugate,  angles. 
What  do  all  these  sums  equal  ?  What  inference  can  you 
draw  as  to  the  sum  of  a  pair  of  exterior,  interior,  or  conju- 
gate angles  of  two  parallels  crossed  by  a  transversal  ? 

12.  If  one  of  two  adjacent  angles  is  94°,  116°,  108°,  or 
75°,  how  many  degrees  is  the  other  ? 

13.  If  the  adjacent  angle  of  an  alternate  interior  angle  is 
70°,  104°,  60°,  or  45°,  how  many  degrees  are  the  alternate 
interior  angles  ? 


/  \  /  /•; /.' 8 Ei ' TED    PARALLELS.  15 

14.  If  tlic  adjacent  angle  of  an  alternate  exterior  angle  is 
QO  ,94  .  L28  .  or  165°,  how  many  degrees  are  the  alternate 

:  ior  angl< 

15.  The  adjacent  angle  ol  an  interior  angle  is  52  .  '■>■>  . 

64°,  or  L'-S  ;  how  many  degrees  in  each  of  the  interior  angles  ? 

16.  The  adjacent  angle  Of  an  exterior  angle  is  SN  .  1L'<>  . 
L09  .  mi-  155°;  how  many  degrees  in  each  of  the  exterior 
angl< 

17.  The  adjacent  angle  of  a  conjugate  angle  is  15°,  32°, 
50°,  or  82°;  what  is  the  measure  of  each  of  the  conjugate 
angl< 

18.  If  one  of  a  pair  of  conjugate  angles  is  58°,  95°,  1 12°, 
or  1.-.U \  what  is  the  other?      Why1.' 

19.  If  one  of  a  pair  of  opposite  angles  is  80°,  120°,  4G°, 
Or  160°,  what  is  the  other?     How  can  yon  tell? 

20.  Draw  a  transversa]  across  two  lines  that  are  not 
parallel,  and  indicate  the  angles  formed  by  letters  or  fig- 
ures. Compare  the  measures  of  the  corresponding  angles 
on  the  diverging  side;  also  those  on  the  converging  side. 
Compare  the  alternate  interior  angles;  also  the  alternate 
exterior  angles, 

21.  With  the  opening  of  the  dividers  (radius)  used  in 
comparing  the  foregoing  angles,  describe  a  circle.  Lay  off 
upon  its  circumference  the  sum  of  the  interior  angles  on 
the  diverging  side  of  the  transversal,  and  compare  it  with 
two  right  angles.  In  the  same  manner  compare  with  two 
right  angles  the  sum  of  the  interior  angles  on  the  con- 
verging side,  that  of  the  exterior  angles  on  the  diverging 

ihe  exterior  angles  on  the  converging  side,  and  the 
conjugate  angles. 

22.  By  what  relation  of  angles  can  you  determine  whether 
two  lines  crossed  by  a  transversal  are  parallel?  How  could 
yon  determine  if  they  were  not  crossed  by  a  transversal? 


16  CONSTRUCTIVE  QEOMETBT. 

23.  Could  you,  from  a  point  without  a  line,  by  means  of 
angles,  draw  a  parallel  to  the  line  ?  Explain  how.  Would 
a  transversal  help  you  to  do  it  ?     How  ? 


IV.    TRIANGLES. 
1.   The  Angles  of  Triangles. 

1.  Draw  a  triangle,  and  on  the  circumference  of  a  circle 
determine  the  sum  of  its  angles. 

2.  If  one  of  the  angles  of  a  triangle  were  enlarged,  Avhat 
effect  would  the  change  have  upon  the  other  angles  ? 

3.  If  two  of  its  angles  were  enlarged,  how  would  it 
affect  the  third?  In  either  case,  would  the  sum  of  the 
angles  be  changed?  What  general  conclusion  may  there- 
fore be  drawn  concerning  the  sum  of  the  three  angles  of 
a  triangle? 

4.  If  two  of  the  angles  of  a  triangle  are  65°  and  52°,  27° 
and  74°,  or  85°  and  89°,  what  is  the  third  angle  ? 

5.  How  many  angles  of  90°  (right  angles)  can  a  triangle 
contain  ?  Draw  such  a  triangle,  and  designate  it  abc,  by 
writing  one  of  these  letters  at  the  vertex  of  each  angle. 
Which  sides  include  the  right  angle  ?     How  do  these  sides 

'meet  each  other  ? 

6.  Name  the  side  opposite  the  right  angle.  This  side 
is  the  hypotenuse. 

7.  Which  is  the  longest  side  in  every  right-angled 
triangle  ? 

8.  What  is  the  sum  of  the  angles  at  the  hypotenuse  ? 
What  kind  of  angle  is  each  of  them  ? 

9.  How  many  degrees  in  one  of  the  angles  at  the  hy- 
potenuse; when  the  other  is  29°,  40°,  67°,  or  71°? 


TRIANGLES.  17 

10.  How  many  obtuse  angles  can  a  triangle  have? 

11.  Draw  a  triangle  containing  an  obtuse  angle. 

12.  In  such  a  triangle,  which  of  the  sides,  according  to 
h,  lies  opposite  the  obtuse  angle  ? 

13.  How  does  the  sum  of  the  angles  at  the  longest  side 
compare  with  *.)<)  '/  What,  kind  of  angle,  therefore,  is  each 
of  them  ? 

14.  If  the  obtuse  angle  and  one  of  the  acute  angles  are 
115  and  --  .  l"l  and  35°,  or  129°  and  31°,  what  is  the 
third  angle  ? 

15.  From  a  point  without  a  line,  draw  a  perpendicular 
to  the  line.  From  the  same  point,  and  on  the  same  side  of 
the  perpendicular,  draw  also  several  oblique  lines  to  the 
line. 

16.  What  kind  of  a  triangle  do  the  perpendicular  and 
nearest  oblique  line  form  ? 

17.  How  does  the  oblique  line  compare  in  length  with 
the  perpendicular  ?    According  to  what  inference  ? 

18.  What  kind  of  triangle  is  that  formed  by  the  oblique 

lines? 

19.  How  do  the  oblique  lines  compare  with  each  other 
in  length  ?     According  to  what  inference  ? 

20.  How  many  perpendiculars  can  be  drawn  from  a  point 
without  a  line  to  the  line? 

21.  How  many  equal  oblique  lines  can  be  drawn  from 
a  point  without  a  line  to  the  line?  How  many,  from  the 
same  point,  on  one  side  of  a  perpendicular  ? 

22.  From  a  point  without  a  line,  draw  two  equal  oblique 
lines  to  the  line,  and  compare  their  distances  from  the  foot 
of  the  perpendicular  to  the  line.     Compare  also  the  distances 


18  CONSTRUCTIVE  GEOMETRY. 

of  two  unequal  oblique  lines  from   the   foot   of   the   per- 
pendicular. 

23.  What  general  inference  can  you  draw  concerning  the 
distances  of  equal  and  of  unequal  lines  from  the  foot  of 
the  perpendicular  ? 

24.  How  many  angles  less  than  90°  (acute  angles)  can 
a  triangle  have  ?     Draw  such  a  triangle. 

25.  Two  angles  of  an  acute-angled  triangle  are  49°  and 
51°,  53°  and  69°,  or  80°  and  79°;  what  is  the  third  angle  ? 

26.  How  many  angles  of  a  triangle  may  contain  more 
than  180°,  or  be  convex  ? 

27.  Draw  a  triangle.  What  is  the  sum  of  the  interior 
and  exterior  angles  at  each  corner  ?  What,  of  all  the 
corners  ?  How  much  of  this  sum  is  derived  from  the 
interior  angles  ?     How  much,  from  the  exterior  ? 

28.  What  inference  can  be  drawn  concerning  the  sum 
of  the  exterior  angles  of  a  triangle  ? 

29.  Draw  a  triangle,  and  prolong  one  of  its  sides  beyond 
the  vertex.  The  angle  thus  formed  without  the  triangle  is 
called  an  exterior  (outer)  angle.  Find  the  difference  between 
the  exterior  angle  and  the  sum  of  the  opposite  interior 
angles. 

30.  What  inference  can  you  draw  concerning  the  exterior 
angle  of  a  triangle  when  compared  with  the  sum  of  its  two 
opposite  interior  angles  ? 

31.  The  two  opposite  interior  angles  of  a  triangle  are  70° 
and  65°,  43°  and  82°,  or  46°  and  54°;  what  is  the  exterior 
angle  ? 

32.  The  exterior  angle  is  134°  and  one  of  the  opposite 
interior  angles  52°,  84°,  56°,  or  41° ;  what  is  the  other  ? 


TRIANGLES.  19 

2. —  The  Sides  of  Triangles. 
a.  —  Triangles  of  Unequal  Sides  (Scalene  Triangles). 

1.  Construct  a  triangle  of  three  lines  of  which  two 
together  shall  be  shorter  than  tin*  third.  Do  the  none 
with  three  lines  of  which  two  together  shall  be  equal  to 
tin-  third  ;  also  with  three  lines  of  which  two  together  shall 
be  greater  than  the  third.  What  inference  can  you  draw 
concerning  the  relation  of  the  sum  of  any  two  sides  of  a 
triangle,  when  compared  with  the  third  side? 

2.  Draw  a  triangle  of  which  no  two  sides  shall  be  equal, 
and  compare  the  angles.  Are  any  of  them  equal?  Which 
one  (according  to  size)  is  found  opposite  the  longest  side  ? 
Which,  opposite  the  next  longest?  Which,  opposite  the 
shortest  ?  What  inference  can  you  therefore  draw  concern- 
ing the  relation  of  the  angles  and  their  opposite  sides,  of  a 
triangle  '.' 

b. —  Triangles  of  Two  Equal  Sides  (Isosceles  Triangles). 

1.  Draw  a  triangle  having  two  equal  sides.  The  point  in 
which  the  equal  sides  meet  is  the  vertex  of  the  triangle, 
and  the  side  opposite  the  vertex  is  the  base. 

2.  At  which  side  do  you  find  the  angles  that  lie  opposite 
the  equal  sides?  Compare  the  measures  of  these  angles. 
What  is  the  inference  concerning  the  angles  at  the  base  of 
an  i sosceles  triangle  ? 

3.  An  angle  at  the  base  is  48°,  63°,  27°,  or  77°;  what  is 
the  measure  of  the  angle  at  the  vertex? 

4.  The  angle  at  the  vertex  is  96°,  78°,  44°,  or  109°;  what 
is  each  of  the  angles  at  the  base  ? 

5.  Prom  the  vertex  draw  a  line  to  the  middle  of  the  base, 
and  compare  the  parts  into  which  it  divides  the  angle  at 


20  CONSTRUCTIVE  GEOMETRY. 

the  vertex.  Compare  also  the  adjacent  angles  which  it 
forms  with  the  base.  What  kind  of  angles  are  they  and 
how  does  the  line  meet  the  base  ?  Into  what  kind  of  tri- 
angles does  it  divide  the  isosceles  triangle  ? 

6.  Can  you  erect  a  perpendicular  to  the  line  ab  at  the  point 
c,  using  no  instruments  but  a  pair  of  compasses  and  a  straujl it- 
edge  or  ruler  f  If  you  should  mark  a  point  at  equal  dis- 
tances on  each  side  of  c,  and  upon  the  line  connecting  the 
two  points,  as  a  base,  construct  an  isosceles  triangle,  could 
you  with  its  aid  erect  the  perpendicular?  Would  a  line 
from  c  to  its  vertex  be  the  perpendicular  ?  According  to 
what  inference  ? 

7.  From  a  point  c,  without  a  line  ab,  draw  a  perpendicular 
to  the  line,  using  the  same  instruments  as  in  previous  case. 
Would  the  isosceles  triangle  assist  you  in  doing  it  ?  If  so, 
explain  how. 

8.  With  the  same  instruments  as  before  and  the  isosceles 
triangle,  divide  a  line  into  two  equal  parts.  Can  you  do  it 
without  the  triangle  ?     If  so,  how  ? 

9.  By  means  of  the  isosceles  triangle  and  the  instru- 
ments used  before,  divide  an  angle  into  two  equal  parts.  If 
you  can  do  it  without  the  triangle,  do  so. 

c.  —  Triangles  of  Three  Equal  Sides  (Equilateral  Triangles). 

1.  Construct  a  triangle  of  three  equal  sides.  Measure 
the  angles.  How  do  they  compare  in  size  ?  Each  one  is 
what  part  of  a  right  angle  ?     How  many  degrees  in  each  ? 

2.  At  the  end  of  a  line,  to  erect  a  perpendicular  to  the 
line.  —  Let  ab  be  the  line  and  a  the  point  at  which  the  per- 
pendicular is  to  be  erected.  From  a  towards  b  lay  off  an 
assumed  distance  to  c,  as  the  base  of  an  equilateral  triangle, 
and  construct  the  triangle.      Mark  the  vertex  of  the  tri- 


TRIANGLB&  21 

angle  d.     Prolong  od  to  s,  making  de  equal  to  cd,  and  draw 
a  line  from  e  to  a. 

3.  What  kind  of  triangle  is  arte?  How  many  degrees  in 
the  angle  ead?  En  ode?  In  aed?  In  eac?  How  do  the 
lines  ae  and  ab  meet  each  other  '.' 

(I.  —  The  Congruence,  or  Agreement,  of  Triangles. 

1.  Constrnei  two  oneqnal-sided  (scalene)  triangles,  mak- 
ing the  sides  of  one  of  the  same  Length  as  the  correspond- 
ing sides  of  the  other.  Compare  the  angles  opposite  the 
longest  sides  of  each,  those  opposite  the  next  longest  sides, 
and  those  opposite  the  shortest  sides.  What  have  you 
found  about  the  angles  lying  opposite  equal  sides?  If  you 
should  cut  one  of  the  triangles  out,  and  lay  it  upon  the 
other,  in  what  respects  would  they  agree?  If  two  triangles 
agree  in  the  lengths  of  their  sides,  what  may  be  said  of  the 
triangles  ?     (They  are  congruent  —  agree  in  all  like  respects.) 

2.  Construct  two  equal-sided  angles,  and  connect  the  ends 
of  the  sides  of  each.  You  have  thus  formed  two  triangles 
of  which  two  sides  and  the  included  angle  of  one  are  equal 
to  two  sides  and  the  included  angle  of  the  other.  Compare 
the  remaining  sides  and  angles.  What  may  therefore  be 
said  of  two  triangles  of  which  two  sides  and  the  included 
angle  of  one  are  equal  to  two  sides  and  the  included  angle 
of  the  other? 

3.  Construct  two  triangles  of  which  two  angles  and  the 
included  side  of  one  shall  be  equal  to  two  angles  and  the 
included  side  of  the  other,  and  compare  the  remaining  like 
parts.  What  inference  may  be  drawn  concerning  two  such 
triangles  ? 

4.  Construct  a  triangle  of  which  two  sides  shall  be  of 
unequal  length.     Construct  an  angle  equal  to  that  opposite 


22  CONSTRUCTIVE  GEOMETRY. 

the  longer  of  the  two  sides  of  the  triangle,  and  make  one 
of  its  sides  eqnal  to  the  shorter  of  the  two  sides.  With 
the  end  of  the  shorter  side  as  a  center,  and  the  longer  side 
as  a  radius,  describe  a  curve  (arc  of  circle)  which  shall 
intersect  the  third  side.  The  triangle  thus  constructed  is 
equal  to  the  other  in  two  of  its  sides  and  the  angle  opposite 
the  greater  side.  Compare  the  measures  of  the  remaining 
similar  parts.  What  inference  may  be  drawn  concerning 
two  such  triangles? 

V.  FOUR-SIDED  FIGURES,   OR  QUADRI- 
LATERALS. 

1..  Construct  a  quadrilateral  (four-sided  figure).  Lay  off 
its  angles  successively  on  the  arc  of  a  circle,  and  find  the 
number  of  degrees  in  their  sum.  What  general  inference 
may  be  drawn  of  the  sum  of  the  angles  of  a  quadrilateral  ? 

2.  If  three  angles  of  a  quadrilateral  are  102°-78°-96°, 
59°-120°-92°,  or  67o-117°-101°,  what  is  the  fourth  angle  ? 

3.  How  many  angles  of  90°  can  a  quadrilateral  have  ? 
Construct  such  a  quadrilateral. 

4.  How  many  angles  greater  than  90°  (obtuse  angles) 
can  a  quadrilateral  have  ?     Construct  such  a  figure. 

5.  How  many  acute  angles  can  a  quadrilateral  have  ? 

6.  If  three  angles  of  a  quadrilateral  are  89°-790-69°, 
what  kind  of  angle  is  the  fourth  ?  If  three  of  the  angles 
are  55°-45°-35°,  what  kind  of  angle  is  the  fourth  ?  Con- 
struct each  of  the  quadrilaterals,  and,  of  the  second,  con- 
struct first  the  convex  angle. 

7.  How  many  right  angles  in  the  sum  of  the  interior 
and  exterior  angles  of  a  quadrilateral  ?  How  many  in  the 
exterior  angles  ? 


qUAbini.ATERALS.  23 

8.  Intersect  (cross)  two  parallel  lines  by  two  parallel 
lines.  What  relation,  as  to  direction,  do  the  opposite  sides 
Off  th*'  quadrilateral  thus  formed  bear  to  each  other?  Such 
a  quadrilateral  is  called  a  parallelogram, 

9.  Intersect  two  parallel  lines  by  two  others  that  are 
not  parallel.  How  many  of  the  sides  of  the  quadrilat- 
eral formed  arc  parallel?  Such  a  quadrilateral  is  called 
a  trapezoid, 

10.  If  two  converging  or  diverging  lines  are  crossed  by 
two  other  lines  that  are  not  parallel,  the  included  quadri- 
lateral is  called  a  trapezium,  of  which  no  sides  are  parallel. 

1.     Till:     1' A  liALLELOGRAM. 

1.  Construct  a  parallelogram  by  crossing,  or  intersecting, 
two  parallel  lines  by  two  other  parallels. 

.  — Two  angles  that  lie  at  the  ends  of  the  same  side  of  ■  par- 
allelogram are  called  adjacent  angle*,  and  the  others,  opposite  any les. 

2.  What  can  you  say  of  the  sum  of  two  adjacent  angles 
of  a  parallelogram?  According  to  what  inference  of  par- 
allels crossed  by  a  transversal  ? 

3.  How  do  the  opposite  angles  of  a  parallelogram  com- 
pare in  size? 

4.  If  an  angle  of  a  parallelogram  is  94°,  72°,  or  64°  (an 
oblique  angle),  how  large,  and  of  what  kind,  are  the  remain- 
ing angles  ?  If  one  of  the  angles  is  90°,  how  large,  and  of 
what  kind,  are  the  remaining  angles  ? 

l!iMu:K.  — A  parallelogram,  all  of  whose  angles  are  right  angles, 
la  a  right-angled  parallelogram  or  rectangle,  and  one  whose  angles  are 
ObUqofl  is  an  «»/./  d  | H I rallelogram  or  rhomh 

5.  Compare  the  lengths  of  the  opposite  sides  of  a  par- 
allelogram. What  general  inference  may  be  drawn  from 
this? 


24  CONSTRUCTIVE  GEOMETRY. 

6.  Could  you  construct  a  parallelogram,  if  you  had  two 
of  its  sides  and  the  included  angle  given  ?  Show  how  you 
would  do  it. 

7.  What  kind  of  figure  is  a  quadrilateral  whose  opposite 
sides  are  equal  ? 

Note.  —  A  line  connecting  the  opposite  angles  of  a  parallelogram 
is  called  a  diagonal. 

8.  Draw  the  two  diagonals  of  a  parallelogram,  and  find 
into  what  kind  of  parts  they  divide  each  other.  What 
general  inference  can  be  drawn  from  this  division  ? 

9.  A.  right-angled  equilateral  (equal-sided)  parallelogram 
is  called  a  square.  Construct  such  a  parallelogram.  Con- 
struct another,  whose  sides  shall  be  equal  to  those  of  the 
first.  Of  what  parts  of  the  first  did  you  take  the  measure 
to  make  the  second  ?  Of  how  many  sides  of  a  square  is  it 
necessary  to  have  the  length,  in  order  to  construct  the 
square  ?     Why  ? 

Note.  —  A  right-angled  parallelogram  whose  opposite  sides  are 
equal  is  called  a  rectangle  or  oblong. 

10.  Construct  a  rectangle.  Construct  another,  whose 
sides  shall  be  equal  to  those  of  the  first.  Of  how  many 
of  its  sides  is  it  necessary  to  have  the  length,  in  order  to 
construct  a  rectangle  ?     Why  ? 

Note.  —  An  oblique-angled  equilateral  parallelogram  is  called  a 
rhombus. 

11.  Construct  a  rhombus.  Construct  another,  whose  sides 
and  angles  shall  be  equal  to  those  of  the  first.  What  parts 
of  the  first  did  you  measure  to  construct  the  second  ? 

12.  What  must  be  known  in  order  to  construct  a  rhombus 
of  given  form  and  dimensions  ? 


QUADRILATERALS.  25 

13.  Construct  a  rhomboid  Construct  another,  whose 
corresponding  sides  and  angles  shall  be  equal  to  those  of 

tin-  first. 

14.  What  nnisl  !»•' known  in  order  to  construct  a  rhomboid 

■  in  form  ami  dimensions'.' 

15.  In  what  parallelograms  are  the  diagonals  of  the  same 
length  '.'     In  what,  of  unequal  length? 

16.  In  what  parallelograms  are  the  adjacent  angles  formed 
by  the  two  diagonals  equal,  and  the  diagonals  perpendicular 
to  each  other  ? 

2.   The   Symmetrical   Trapezoid. 

Note.  — If  two  parallels  be  crossed  by  two  converging  lines,  so  that 
the  mgtafl  made  with  the  same  parallel  are  equal,  the  resulting  figure 
is  a  symmetrical  trapezoid. 

1.  Construct  a  symmetrical  trapezoid,  and  compare  the 
lengths  (1)  of  the  parallel  sides  and  (2)  of  the  converg- 
ing sides.  What  inference  do  you  draw  concerning  the 
lengths  of  the  opposite  sides  of  such  a  trapezoid? 

2.  What  kind  of  angles  are  those  on  the  longer  parallel  ? 
Those  on  the  shorter? 

3.  What  relation  do  the  angles  on  either  of  the  oblique 
linos  bear  to  each  other  ?  What  is  their  sum  ?  According 
to  what  inference  ? 

4.  How  do  the  angles  at  each  of  the  parallels  compare 
with  each  other  in  size?  If  one  of  the  angles  is  47°,  52°, 
80°,  or  28\  how  large  is  each  of  the  others  ? 

5.  I  )ra  w  the  diagonals  of  the  trapezoid,  and  compare  their 
lengths,     What  do  you  find  '.' 

6.  Compare  the  parts  of  tho  diagonals  ihat  meet  the 
longer  parallel ;  also  those  that  meet  the  shorter.     Are  all  of 


26  CONSTRUCTIVE  GEOMETRY. 

them  of  the  same  length  ?     If  not,  what  difference  do  you 
find? 

7.  Draw  perpendiculars  from  the  ends  of  the  shorter  par- 
allel to  the  longer,  and  compare  the  parts  cut  off  from  the 
longer.     How  do  their  lengths  compare  ? 

8.  Lay  off  the  length  of  the  shorter  of  the  two  parallels 
of  a  symmetrical  trapezoid  upon  the  longer,  to  find  their 
difference.  From  the  point  thus  found  draw  a  line  which, 
with  the  difference  and  its  adjoining  oblique  line,  shall 
form  a  triangle:  What  kind  of  triangle,  with  regard  to  its 
sides,  is  it  ?  What  relation  does  the  sum  of  its  sides  bear 
to  the  base  ?  According  to  what  inference  ?  What  rela- 
tion does  either  of  the  oblique  sides  bear  to  half  the  base  ? 
What  kind  of  quadrilateral  is  that  adjacent  to  the  triangle  ? 

9.  Given  three  lines,  two  of  them  to  be  the  parallels,  and  the 
third  one  of  the  oblique  sides,  to  construct  a  symmetrical  trape- 
zoid.—  Measure  off  on  the  longer  of  the  parallels  the  length 
of  the  shorter.  Upon  the  remainder,  or  difference,  as  the 
base,  and  the  third  line  as  one  of  the  sides,  erect  an  isosceles 
triangle.  Adjacent  to  the  triangle,  construct  a  parallelo- 
gram, two  of  whose  sides  shall  be  respectively  the  part  meas- 
ured off  on  the  longer  parallel  and  the  adjoining  oblique 
side  of  the  triangle.  The  parallelogram  and  the  triangle 
together  form  the  trapezoid.  Explain  why.  How  many 
sides  of  a  symmetrical  trapezoid  must  be  given  to  construct 
the  trapezoid  ? 

10.  Given  an  acute  angle  and  two  lines,  one  of  the  lines  to 
serve  as  the  measure  of  the  shorter  parallel,  the  other  as  that  of 
one  of  the  oblique  sides,  of  a  symmetrical  trapezoid',  to  construct 
the  trapezoid. — Make  one  of  the  sides  of  the  angle  equal  to 
the  oblique  line,  and,  with  it  as  a  radius  and  its  end  as  a 
center,  describe  an  arc  that  shall  intersect  the  other  side. 
Connect  the  intersection  with  the  end  used  as  center  of  the 


THE  POLYGON,  27 

arc.  What  kind  of  triangle  have  you  formed  '!  Why  ?  To 
the  base  of  the  triangle  add  the  parallel,  and  upon  the  pari 

thus  added  erect  a  parallelogram  in  the  same  manner  as  in 
the  previous  case.  What  do  the  two  figures  together  form  ? 
Why  '.'  How  many  elements  (conditions  or  parts)  of  asym- 
metrical trapezoid  are  necessary  to  construct  the  trapezoid  ? 

11.  If  instead  of  an  acute  angle  an  obtuse  angle  were 
given,  how  would  you  find  one  of  the  acute  angles  ? 

12.  If  you  had  given  the  two  parallel  sides  and  one  of 
the  acute  angles,  as  elements  of  a  symmetrical  trapezoid, 
how  would  you,  with  the  aid  of  the  isosceles  triangle,  con- 
struct the  trapezoid  ? 

VI.  THE  POLYGON. 

1.  A  figure  that  has  more  than  four  corners  or  angles  is 
called  a  polygon  (many  corners).  Triangles  are  sometimes 
called  polygons. 

2.  Construct  a  five-sided  figure  (pentagon),  and  from  a 
point  within  draw  lines  to  all  the  corners.  Into  how  many 
parts  is  the  polygon  divided  ?  What  kind  of  figures  are  the 
parts  ?  How  does  the  number  of  sides  of  the  polygon  com- 
pare with  the  number  of  parts  ?  How  does  the  number  of 
triangles  compare  with  the  number  of  corners  of  the  poly- 
gon ?  Would  the  same  hold  good  in  any  polygon  ?  Try 
whether  it  would. 

3.  I  low  many  right  angles  does  the  sum  of  all  the  angles 
of  the  five  triangles  of  the  foregoing  polygon  equal  ?  How 
does  the  number  of  right  angles  compare  with  the  number 
of  corners  of  the  pentagon  ?  What  angles  of  the  triangles 
do  not  constitute  part  of  the  corners  of  the  pentagon  ?  What 
is  their  sum  ?  What  is  the  sum  of  the  other  angles  of  the 
triangles,  those  that  constitute  the  angles  of  the  pentagon? 


28  CONSTRUCTIVE  GEOMETRY. 

How  does  this  sum  compare  with  the  sum  of  all  the  angles 
of  the  triangles  ? 

4.  If  you  had  given  the  number  of  corners  of  a  polygon, 
could  you  find  the  number  of  right  angles  in  its  interior 
angles  ?     How  ? 

5.  Suppose  a  figure  has  8,  12,  15,  or  18  sides,  how  many 
right  angles  are  its  interior  angles  equal  to  ? 

6.  How  many  right  angles  in  the  interior  and  exterior 
angles  of  a  pentagon  ?  What  is  the  sum  of  the  exterior 
angles  ?  How  does  this  sum  compare  with  the  number  of 
corners  of  the  figure  ? 

7.  What  is  the  sum  of  the  exterior  angles  of  a  polygon 
having  6,  8,  10,  or  16  corners  ? 

8.  With  any  suitable  radius  describe  a  circle,  and  find 
how  many  times  the  radius  can  be  applied  to  the  circum- 
ference. Connect  the  points  with  straight  lines.  What 
kind  of  polygon  have  you  formed  ?  How  many  more  sides 
has  it  than  angles  ?  When  all  the  sides  and  all  the  angles 
of  a  polygon  are  equal,  the  figure  is  called  a  regular  polygon. 

9.  Construct  a  regular  twelve-cornered  figure  (dodecagon) 
by  dividing  each  of  the  arcs  of  a  six-cornered  figure  (hexagon) 
into  halves,  and  connecting  the  adjacent  points  with  straight 
lines. 

10.  A  straight  line  that  passes  through  the  center  of  a 
circle  and  intersects  the  circumference  in  two  points  is  a 
diameter  of  the  circle. 

11.  If  you  should  draw  two  diameters  of  a  circle  at  right 
angles  to  each  other,  and  connect  the  adjacent  ends  by 
straight  lines,  what  kind  of  figure  would  you  form  ?  Draw 
such  a  figure.  Divide  its  arcs  into  halves,  and  construct 
the  resulting  figure ;  again  divide  the  latter  into  halves,  and 
construct  the  corresponding  figure. 


A  HE  AS   OF  FIGURES.  29 

12.  How  many  degrees  in  the  interior  angle  of  a  regular 
5,  <•.  8,  10,  or  15  sided  polygon? 

13.  From  the  center  of  a  regular  polygon  draw  lines  to 
tin-  corners  and  compare  their  lengths.  In  what  respects 
arc  tin-  resulting  triangles  equal?  What  do  you  conclude 
bom  this?  According  to  what  inference?  How  do  the 
angles  at  the  center  compare  with  each  other?  How  many 
degrees  in  an  angle  at  the  center  of  a  regular  G,  8,  9,  15,  oi- 
ls cornered  figure? 

VII.    THE  AREAS   OP   FIGURES   INCLOSED   BY 
STRAIGHT   LINES. 

1.    Tin:   E&BOl angle,  or  Right-angled   Parallelogram. 

1.  Construct  a  rectangle  whose  sides  shall  be  G  inches  and 
4  inches  in  length,  or,  if  drawn  upon  the  blackboard,  larger, 
—  say  9  inches  and  5  inches  respectively.  Since  in  a  paral- 
lelogram any  side  may  be  regarded  as  the  base,  the  longer 
side  (G  inches)  may  be  taken  for  it ;  and  since  in  a  parallelo- 
gram, also,  a  perpendicular  from  one  side  upon  the  other, 
or  upon  its  prolongation,  may  be  taken  as  the  height,  the 
shorter  side  (4  inches)  may  in  this  case  be  taken  for  it. 
1  )ivide  the  sides  into  inches,  and  connect  the  opposite  points 
by  straight  lines. 

2.  Into  what  kind  of  parallelograms  is  the  rectangle 
divided".'  How  long  is  each  side  of  one  of  the  parallelo- 
grams ? 

3.  A  figure  of  four  equal  sides  and  with  all  its  angles 
right  angles  is  called  a  square.  When  the  sides  of  a  square 
are  an  inch  in  length,  the  figure  is  called  a  square  Inch  ;  when 
a  foot  in  length,  a  square  foot;  when  a  yard,  a  square  yard, 

etc. 

• 

4.  In  the  foregoing  rectangle  (6  by  4),  (a)  How  many 


30  CONSTRUCTIVE  GEOMETRY. 

of  the  squares  lie  in  the  first  row,  or  the  row  on  the  base  ? 
(&)  How  many,  in  the  second  row  ?  (c)  How  many,  in  the 
third  ?  (d)  How  does  the  number  in  each  row  compare  with 
the  number  of  inches  in  the  base  ?  (e)  How  does  the  num- 
ber of  rows  compare  with  the  number  of  inches  in  width  or 
height?  (/)  If  we  should  multiply  the  number  in  the  first 
row  by  the  number  of  rows,  how  would  the  product  differ 
from  the  number  of  squares  in  the  whole  figure  ?  (g)  If  we 
should  multiply  the  number  of  inches  in  the  length  of  the 
figure  (the  base)  by  the  number  in  its  width  or  height,  how 
would  the  product  differ  from  the  last  found  product,  or 
area,  of  the  figure  ?  (h)  How,  therefore,  may  the  area  of  a 
parallelogram  or  rectangle  be  found  without  dividing  the 
figure  into  squares  ? 

Note.  —  The  numbers  in  most  of  the  problems  and  exercises  that 
follow  are  based  upon  the  metric  system  ;  but  the  pupils  may,  if  they 
prefer  it,  regard  them  as  inches,  feet,  or  other  dimensions,  —  linear, 
square,  etc.,  as  the  case  may  be. 

The  teacher  or  pupil  who  prefers  to  change  the  given  numbers  to 
those  of  the  common  system  can  easily  do  so  by  bearing  in  mind  the 
following  relations : 

The  meter  =  39.37  inches. 

The  decimeter  (^  of  meter)      =  3.937  inches. 

The  centimeter  (TiQ  of  meter)  =  .3937  inches. 

The  square  meter  =  1550  square  inches. 

The  cubic  meter  =  35.316  cubic  feet. 

5.  Find  the  areas  of  the  following  right-angled  parallelo- 
grams, the  base  and  the  height  of  each  being  given. 

Note. — in.  stands  for  meter,  cm.  for  centimeter,  dm.  for  deci- 
meter, sq.  cm.  for  square  centimeter,  and  cu.  dm.  for  cubic  decimeter. 

Base  Height.  Base.  Height. 

(1)  13  cm.  8  cm.  (4)  17  cm.  9  cm. 

(2)  24  cm.  6  cm.  (5)  12  cm.  11  cm. 

(3)  15  cm.  9  cm.  (6)  14  cm.  17  cm. 

6.  Find  the  bases  of  the  following  rectangles,  the  area  and 
the  height  being  given : 


AREAS  OF  FIGURES.  31 

Area.  Il:-ht.  Area.  Height. 

(1)     94  sq.  cm.        7  cm.  (4)     127  sq.  cm.         13  cm. 

(2;  106  sq.  cm.         9  cm.  (5)       87  sq.  cm.  9  cm. 

(0)  70  sq.  cm.        8  cm.  (0)     116  sq.  cm.         1 1  cm. 

7.    Find  the  heights  of  the  following  rectangles,  the  area 
and  tin'  base  being  given: 

BtM.  Bam. 

(1)  124  sq.  cm.        31cm.  (4)     91  sq.  cm.  13  cm. 

(2)  144  sq.  cm.        24  cm.  (5)   108  sq.  cm.  18  cm. 

(3)  96  sq.  cm.        12  cm.  (6)  119  sq.  cm.  17  cm. 

8.  Since  both  dimensions  of  a  square  are  the  same,  how 
would  you  find  its  area  or  surface? 

9.  The  side  of  a  square  is  9,  G,  11,  or  15 ;  what  is  its  area  ? 

10.    When  the  surface  of  a  square  is  144,  81,  169,  225,  or 
301,  what  is  its  side  *.' 

2.   The  Squawk. 

1.  Construct  a  square  and  prolong  its   base   ab   to  c, 

making  be  less  than  ab. 

2.  Upon  ac  construct  a  second  square  which  shall  in- 
clude the  square  upon  ab. 

3.  Prolong  the  sides  of  the  square  ab  to  meet  those  of  ac. 
The  square  upon  ab  is  written  ab  ;  that  upon  ac,  ac . 

4.  We  must  now  find  the  length  of  ac  or  ab  +  be. 

5.  When  aV  is  100,  400,  900,  1600,  2500,  3600,  4900, 
0400,  or  8100,  how  long  is  ab  ? 

6.  After  subtracting  ab  from  ac ,  what  figures  remain  ? 

Note.  —  The  pupils  should  be  led  to  see  that  both  of  the  rectangles 
main  are  together  equal  to  a  parallelogram  whose  base  is  twice 
oft,  and  its  height  be. 

7.  With  what  numbers  must  the  remainder  be  divided 
to  obtain  be,  when  ab  Efl  10,  20,  30,  40,  50,  60,  70,  80,  or  90? 


32  CONSTRUCTIVE  GEOMETRY. 

8.  How  many  square  meters,  inches,  or  whatever  the 
denomination  may  be,  remain  for  the  small  square  in  the 
figure  when  be  is  1,  2,  3,  4,  5,  6,  7/8,  or  9  ? 

9.  What  is  the  length  of  the  side  of  a  square  whose  sur 
face  is  4225,  7569,  17G4,  6241,  9025,  3364,  or  5476  ? 

10.  In  the  foregoing  figure,  prolong  ac  to  d,  making  cd 
yet  smaller  than  be. 

11.  Upon  ad  erect  a  third  square  (ad ),  which  shall  in- 
clude ac . 

12.  Prolong  the  sides  of  ac  to  meet  those  of  ad2. 

13.  The  problem  now  is  to  find  the  length  of  ad  or 
ab  +  be  -f-  cd. 

14.  How  long  is  ab,  when  aV  is  10000,  40000,  90000, 
160000,  250000,  360000,  490000,  640000,  or  810000  ? 

15.  Next  find  the  length  of  be.     How  can  you  do  it  ? 

16.  How  many  square  meters  or  inches  (as  the  case  may 
be)  are  in  the  middle  square  (ac ),  where  be  is  10,  20,  30, 
40,  50,  60,  70,  80,  or  90  ? 

17.  If  we  regard  the  sides  of  ac  as  the  base,  and  cd  as 
the  height  or  width  of  the  adjacent  rectangles,  how  can  we 
find  cd  ? 

18.  How  many  square  meters  remain  for  the  smallest 
square  in  the  figure  if  cd  is  1,  2,  3,  4,  5,  6,  7,  8,  or  9  ? 

19.  What  is  the  side  of  a  square  whose  area  is  24336, 
34969,  60516,  47524,  141376,  or  105625  ? 

3.   The  Oblique-angled  Parallelogram. 

1.  Construct  a  rectangle,  and  from  one  end  of  its  base 
draw  an  oblique  line  to  the  side  opposite  the  base.  From 
the  other  end  of  the  base  draw  an  oblique  line  parallel  to 


AREAS  OF  FIGURES.  33 

the  iirst ,  to  meet  the  prolongation  of  the  side  opposite  the 
base. 

2.  What  kind  of  figure  is  the  second  one  drawn  ? 

3.  In  what  respects  do  the  two  figures  agree  f  Both  have 
the  mum  bate  and  lie  between  the  mam  parallel*. 

4.  In  what  respects  do  the  two  triangles — the  one  within 
the  rectangle,  the  other  without  —  agree?  What  may  we 
Conclude  from  this?  According  to  what  inference?  Con- 
grm  i 

5.  How  does  the  area  of  the  interior  triangle  of  the 
rectangle  compare  with  that  of  the  exterior  one?  How, 
therefore,  does  the  area  of  the  oblique-angled  parallelogram 
com  pare  with  that  of  the  rectangle? 

6.  What  factors  must  be  multiplied  together  to  obtain 
the  area  of  either  parallelogram,  or  of  any  parallelogram  ? 

4.   The  Triangle. 

1.  Construct  a  parallelogram  and,  with  a  diagonal,  divide 
it  into  two  triangles. 

2.  In  what  parts  or  respects  do  the  two  triangles  agree? 

3.  Wliat  conclusion  do  you  draw  from  this?  According 
to  what  inference? 

4.  What  part  of  the  parallelogram  is  included  in  each 
triangle  ? 

5.  In  how  many  ways  may  the  area  of  a  triangle  be 
found? 

6.  Find  the  area  of  each  of  the  following  triangles,  the 
and  the  altitude  (height)  being  given  : 

Altitu.l.-  Altitude. 

(1)  8  cm.  9  cm.  (4)     9  cm.  14  cm. 

(2)  12  cm.  8  cm.  (6)  11  cm.  13  cm. 

(3)  16  cm.  7  cm.  (0)   17  cm.  6  cm. 


34  CONSTRUCTIVE  GEOMETRY. 

7.  Find  the  bases  of  the  following  triangles,  the  area  and 
altitude  being  given : 

Area.  Altitude.  Area.  Altitude. 

(1)  144  sq.  cm.        16  cm.  (4)     72  sq.  cm.  9  cm. 

(2)  96  sq.  cm.  8  cm.  (5)  124  sq.  cm.  4  cm. 

(3)  124  sq.  cm.  5  cm.  (6)     84  sq.  cm.        12  cm. 

8.  Find  the  altitudes  of  the  following  triangles,  the  area 
and  base  being  given  : 

Area.  Base.  Area.  Base. 

(1)  78  sq.  cm.    16  cm.       (4)  98  sq.  cm.    7  cm. 

(2)  156  sq.  cm.    20  cm.       (5)  128  sq.  cm.    8  cm. 

(3)  112  sq.  cm.    14  cm.       (6)  81  sq.  cm.    9  cm. 

a.  —  Triangles  of  the  Same  Base  and  Equal  Altitude. 

1.  Construct  a  triangle,  and  through  its  vertex  draw  a 
line  parallel  to  the  base. 

2.  From  various  points  in  the  parallel  draw  lines  to  both 
ends  of  the  base,  thus  also  forming  triangles. 

3.  What  can  you  say  of  the  areas  of  all  these  triangles  ? 
How  are  they  found  ? 

Remark.  —  The  parallel  line  is  called  the  geometrical  locus  of  the 
vertices  of  the  triangles. 

b.  —  The  Right-angled  Triangle. 

1.  Construct  a  right-angled  triangle  whose  sides  shall  be 
to  each  other  in  the  ratio  of  3  to  4, —  say  3  centimeters 
and  4  centimeters,  or  3  inches  and  4  inches,  —  and  find  the 
length  of  the  hypotenuse. 

2.  Upon  the  hypotenuse  and  the  sides  erect  squares. 

3.  Find  the  area  of  each  of  the  squares. 

4.  How  does  the  sum  of  the  areas  of  the  squares  on 
the  two  sides  compare  with  the  area  of  the  square  on  the 
hypotenuse  ?  What  general  inference  may  be  drawn  from 
this? 


AREAS  OF  FIGURES.  35 

5.  If  one  side  of  a  right-angled  triangle  is  5  inches  and 
the  other  1-,  how  long  is  the  hypotenuse? 

6.  What  is  the  sum  of  the  squares  of  the  sides  ? 

7.  What  is  the  square  root  of  the  sum  ?     The  sqiuire  root 

nf  the  sum  is  tltr  trm/th  of  the  hypotenuse. 

8.  If  the  hypotenuse  of  a  right-angled  triangle  is  29 
inches,  and  one  of  the  sides  21  inches,  how  long  is  the  other 
Bide  P 

9.  What  remains  after  subtracting  the  square  of  the 
given  side  from  that  of  the  hypotenuse?  TJie  square  root 
nf  tin  r  ft  the  other  side. 

c.  —  The  Isosceles  Triangle. 

1.  If  in  an  isosceles  triangle  a  line  be  drawn  from  the 
vertex  to  the  middle  of  the  base,  thus  dividing  the  triangle 
into  two  right-angled  triangles,  which  line  of  each  is  the 
hypotenuse? 

2.  Which  lines  are  the  sides  ? 

3.  Find  one  of  the  sides  in  each  of  the  following  isosceles 
triangles,  the  base  and  perpendicular  upon  it  from  the  ver- 
tex being  given : 


.  Rase. 

PtT[«;ndicular. 

Bm* 

Perpendicular. 

(1)  'l2cm. 

9  cin. 

(4)  13  cm. 

5  cm. 

(S      17  rm. 

8  cm. 

(5)  24  cm. 

9  cm. 

5  cm. 

11  cm. 

(6)  18  cm. 

16  cm. 

4.  Find  the  base  of  each  of  the  following  isosceles  tri- 
angles, the  perpendicular  and  one  of  the  sides  of  each 
triangle  being  given: 


Side. 

IVrpendicular. 

Side. 

P.  rj-cndicular. 

(1)  24  cm. 

12  cm. 

(4)  16  cm. 

7  cm. 

(2)  18.1.1. 

11  cm. 

(6)  19  cm. 

8  cm. 

(3)  25  cm. 

9  cm. 

(6)  21  cm. 

13  cm. 

36  CONSTRUCTIVE  GEOMETRY. 

5.   Find   the  perpendiculars   of   the   following   isosceles 
triangles,  the  base  and  one  of  the  sides  being  given : 


Base. 

Side. 

Base. 

Side. 

(1) 

24  cm. 

19  cm. 

(4) 

16  cm. 

24  cm. 

(2) 

17  cm. 

21  cm. 

(5) 

18  cm. 

17  cm. 

(3) 

22  cm. 

18  cm. 

(6) 

12  cm. 

lGcm. 

6.  To  construct  a  square  equal  to  two  given  squares.  —  Con- 
struct a  right-angled  triangle,  whose  sides  shall  be  equal  to 
the  sides  of  the  given  squares.  Upon  the  hypotenuse  as  a 
base,  erect  a  square,  and  this  will  be  equal  to  the  sum  of 
the  given  squares.     Why  ? 

7.  To  construct  a  square  equal  to  the  difference  of  two  given 
squares.  —  Construct  a  right  angle,  making  one  of  its  sides 
equal  to  the  side  of  the  smaller  square.  With  the  end  of 
this  side  as  a  center  and  the  side  of  the  larger  square  as  a 
radius,  describe  an  arc  which  shall  intersect  and  limit  the 
other  side.  A  square  erected  on  the  last-named  side  will 
be  equal  to  the  difference  of  the  given  squares.     Why  ? 

5.    The  Trapezoid. 

1.  Draw  a  horizontal  line  9  inches  long;  4  inches  from  it 
draw  another  7  inches  long  and  parallel  to  the  first.  '  With 
these  two  lines  construct  a  trapezoid,  and  with  a  diagonal 
divide  it  into  two  triangles. 

2.  What  is  the  area  of  the  triangle  that  has  the  longer 
parallel  as  a  base  ?  What,  of  that  which  has  the  shorter  as 
a  base  ? 

3.  What  is  the  area  of  both  triangles,  or  of  the  trapezoid  ? 

4.  If  you  should  multiply  the  sum  of  both  parallels  by 
half  the  distance  between  them,  what  result  would  you 
obtain  ?  Would  it  differ  from  the  area  of  the  trapezoid  ? 
Why? 


AREAS  OF   T1QUB1  37 

5.  Suppose   yOU    should    multiply    the    half   sum    of   the 

parallels  l>y  (he  whole  distance  between  them,  what  result 

would  you  obtain  '.'     Why  *.' 

6.  If  you  should  multiply  the  sum  of  the  parallels  by 
the  distance  between  them,  and  divide  the  product  by  two, 
what  result  would  you  obtain  '.'     Why? 

7.  Find  the  area  of  the  following  trapezoids,  the  Length 
of  the  parallels  and  the  distanee  between  them  being  given: 

Ala,  Dlst.  RttW.                                    Parallels.  Mft  K<-t. 

(1)  16  tod  17  cm.  8  cm.  (4)  18  and  12  cm.  12  cm. 

(2)  23  and  19  cm.  11  cm.  (6)  S7  and  18  cm.  14  cm. 

(3)  34  and  16  cm.  9  cm.  (6)  48  and  34  cm.  16  cm. 

8.  Find  the  distance  between  the  parallels  of  the  fol- 
lowing trapezoids,  the  area  and  the  lengths  of  the  parallels 
being  given : 

Parallels.  Area.  Parallels. 

(1)  96  sq.  cm.       16  and  19  cm.  (4)  186  sq.  cm.      19  and  11  cm. 

_'     1  l-l  sq.  cm.      28  and  20  cm.  (5)  360  sq.  cm.      48  and  24  cm. 

(3)  108  sq.  cm.       13  and  14  cm.  (6)  216  sq.  cm.       12  and  15  cm. 

9.  Find  the  length  of  one  of  the  parallels  of  each  of  the 
following  trapezoids,  the  area,  the  other  parallel,  and  the 
distance  between  the  parallels  being  given : 

•  I.    l>i-t.  IJetw.  Area.  Parallel.    Dist.r..  tw 

1     199  sq.  em.    12  cm.     18  cm.      (4)    88  sq.  cm.      8  cm.     10  cm. 

(2)  288  sq.  cm.    16  cm.     22  cm.       (5)  156  sq.  cm.     12  cm.      14  cm. 

(3)  168  sq.  cm.      8  cm.     31  cm.       (6)  248  sq.  cm.       8  cm.      36  cm. 

10.  Hie  relation  of  the  number  of  sides  of  a  figure  (1)  to 
tin  number  of  <h'<i(fonals,  (2)  to  the  number  of  triangle*  into 
which  ike  figure  may  be  divided. — Construct  a  seven-sided 
figure,  and  from  one  of  its  angles  draw  all  the  possible 
diagonals. 

11.  How  many  more  sides  are  there  than  diagonals? 

12.  How  many  more  sides  are  there  than  yon  formed 
triangles  ? 


38  CONSTRUCTIVE  GEOMETRY. 

13.  What  inference  can  you  draw  from  your  figure  con- 
cerning the  relation  of  the  number  of  sides  pf  a  figure  to 
that  of  its  diagonals  ? 

14.  What  inference  can  you  draw  concerning  the  relation 
of  the  number  of  sides  of  a  figure  to  that  of  the  number  of 
triangles  into  which  the  figure  may  be  divided  ? 

15.  How  many  diagonals  may  be  drawn  from  an  angle, 
and  how  many  triangles  thus  formed,  in  a  figure  of  4,  6,  8, 
10,  12,  or  14  sides  ?     In  a  triangle  ? 

6.   The  Polygon. 

1.  To  find  the  area  of  a  trapezium  and  of  a  polygon, 
divide  the  figure  into  triangles,  and  the  sum  of  their  areas 
or  surfaces  will  be  the  area  of  the  figure. 

2.  Find  the  area  of  each  of  the  following  regular  figures, 
the  number  of  sides,  their  length,  and  the  perpendicular 
upon  them  from  the  center  being  given : 

No.  of  Sides.    Length.           Perp.  No.  of  Sides.    Length.  Perp. 

(1)  5          3.32  m.  2.28  m.  (4)       10        2.50  m.  3.84  m. 

(2)  6            75  cm.       65  cm.  (5)        12  50  cm.  93.3  cm. 

(3)  8            90  cm.  1.08  cm.  .  (6)       16           36  cm.  32  cm. 


VIII.  THE  CIRCLE. 

1.  With  any  assumed  opening  of  the  compasses  describe 
a  circle. 

2.  From  the  middle  point,  or  center,  of  the  circle  draw 
straight  lines,  or  radii,  to  the  circumference  and  compare 
their  lengths. 

3.  What  can  you  say  of  the  lengths  of  the  radii  ? 

4.  Are  any  points  of  the  circumference  farther  from 
the  center  than  others  ?  What,  therefore,  is  the  general 
inference  ? 


THE   CIRC  1. 1  39 

5.  How  does  the  length  of  a  diameter  (through  meas- 
uring) compare  with  thai  of  a  radius? 

6.  How  do  the  different  diameters  compare  with  each 

otli» T  in  length?     What  general  inference  may  therein 
drawn  7 

7.  At  the  end  of  a  radius  erect  a  perpendicular,  and 
notice  in  how  many  points  it  touches  the  circumference. 
Such  a  perpendicular  is  called  a  tangent  (touching  line),  and 
the  point  at  which  it  touches  the  circumference  is  the 
tangent  [><>int. 

8.  At  the  end  of  a  radius  erect  a  line  which  shall  make 
an  oblique  angle  with  the  radius.  In  how  many  points  can 
such  a  line  meet,  or  intersect,  the  circumference?  The 
Dame  of  this  line  is  secant  (cutting  line),  and  the  part 
within  the  circumference  is  a  chord. 

9.  From  the  center  of  a  circle  draw  a  line  to  the  middle 
of  a  chord,  and  compare  the  adjacent  angles  which  it  makes 
with  the  chord.  How  do  the  two  lines  meet  each  other  ? 
Draw  an  inference  from  this. 

10.  If  from  the  center  of  a  circle  a  perpendicular  be 
drawn  to  a  chord,  into  what  kind  of  parts  does  it  divide  the 
chord  ?     What  general  inference  may  be  drawn  from  this  ? 

11.  If  a  perpendicular  be  erected  at  the  middle  of  a 
chord,  through  what  point  in  the  circle  will  it  pass?  What 
inference  may  be  drawn  from  this  ? 

12.  If  two  chords  be  drawn  in  a  circle,  and  a  perpendicu- 
lar be  erected  at  the  middle  of  each,  at  what  point  in  the 
circle  will  the  perpendiculars  meet?  How,  consequently, 
i.iay  the  center  of  a  circle  be  found  ? 

13.  To  draw  a  circumference  through  three  points  not  in 
the  sanu  /;„< .  — Connect  the  points  by  straight  lines,  and  at 
the  middle  of  each  erect  a  perpendicular.     The  intersection 


40  CONSTRUCTIVE  GEOMETRY. 

of  the  perpendiculars  will  be  the  center  of  the  circle,  and 
the  lines  will  be  chords  of  it.  How  many  points,  therefore, 
determine  the  circumference  of  a  circle  ? 

14.  If  you  should  describe  two  circles  that  have  two 
points  in  common,  in  how  many  points  would  they  intersect 
each  other  ? 

15.  In  a  straight  line,  select  three  points  at  suitable  dis- 
tances apart.  With  the  first  as  a  center,  and  the  distance  to 
the  second  as  a  radius,  describe  a  circle.  Likewise,  with 
the  third  as  a  center,  and  the  distance  to  the  second  as  a 
radius,  describe  a  circle.  In  how  many  points  do  the  circles 
touch  ?    Is  the  tangent  point  within  or  without  the  circles  ? 

16.  In  a  straight  line,  select  three  points,  and,  with  the 
first  as  a  center  and  the  distance  to  the  third  as  a  radius, 
describe  a  circle ;  likewise,  with  the  second  as  a  center,  and 
the  distance  to  the  third  as  a  radius,  describe  a  circle.  In 
how  many  points  do  the  circles  touch  each  other  ?  Is  the 
tangent  point  an  interior  or  an  exterior  one  ? 

17.  The  line  in  which  the  centers  and  the  tangent  point 
lie  is  called  the  central  line. 

1.    Peripheral  and  Interior  and  Exterior  Eccentric 

Angles. 

1.  Describe  a  circle,  and  upon  an  arc  of  it  as  a  base  con- 
struct an  angle  whose  vertex  shall  lie  in  the  circumference. 
Such  an  angle  is  called  a  peripheral  angle. 

2.  With  the  radius  of  the  foregoing  circle  and  the  ver- 
tex of  the  angle  as  a  center,  describe  an  arc  that  shall  inter- 
sect both  sides  of  the  angle.  Measure  the  intercepted  arc, 
and  compare  its  length  with  that  upon  which  the  angle 
stands.  How  do  they  compare?  What  inference  may 
therefore  be  drawn  concerning  the  measure  of  a  peripheral 
angle  when  compared  with  a  center  angle  ? 


THE  CIRCLE.  41 

N A  it'jlc  is  one  wbost  v.  rtcx   is  ;tt  the  center,  or 

ifhoM  measuring  are  is  described  with  the  same  radius  as  that  of  tbe 

eircle  Id  which  it  is  formed. 

3.  H<>\\  many  degrees  in  the  arc  intercepted  by  the  sides 
of  a  peripheral  angle  of  24°,  88°,  49°,  57°,  08°,  or  89°? 

4.  How  many  degrees  in  a  peripheral  angle  that  stands 
upon  an  are  of  72°,  9G°,  120°,  144°,  168°,  or  180°? 

5.  Upon  a  straight  line,  to  construct  a  right-angled  triangle, 

• —  Divide  the  given  line  into  two  equal  parts.  With  the 
middle  poinl  us  a  center,  and  half  of  the  line  as  a  radius, 
describe  B  semicircle;  chords  drawn  from  any  point  in  the 
arc  to  the  ends  of  the  given  line  will  form  the  sides  of 
a  right-angled  triangle.  Why  is  the  triangle  thus  formed 
right  angled  ? 

6.  At  any  point  in  a  circumference,  to  draw  a  tangent  to 
eumfen  nee,  —  Describe  a  circumference;  draw  a  radius 

to  the  tangent  point;  then  a  perpendicular  erected  at  its 
extremity,  or  end,  will  be  the  required  tangent. 

7.  From  a  point  without  a  circumference,  to  draw  a  tan- 
geni  to  the  drewmfi rence.  —  From  the  given  point  draw  a 
straight  line  to  the  center  of  the  circle.  Upon  the  line,  as 
a  diameter,  describe  a  second  circle,  and  the  points  in  which 
the  circumferences  intersect  each  other  will  be  the  points  at 
which  the  tangents  will  touch  the  circumference.  To  prove 
it.  draw  radii  to  the  points  of  intersection  in  the  first  circle, 
and  note  the  angles  they  make  with  the  tangents.  What 
inference  can  you  draw  from  a  comparison  of  the  lengths  of 
the  tangents  ? 

8.  In  a  circle,  constnict  an  interior  eccentric  angle  —  an 
angle  whose  vertex  shall  be  neither  in  the  circumference 
nor  at  the  center.  With  the  same  radius  as  that  of  the 
circle,  find  the  measure  of  the  angle,  Prolong  the  sides  of 
the  angle  beyond  the  vertex  to  the  circumference.     With 


42  CONSTRUCTIVE  GEOMETRY. 

the  compasses,  take  the  length  of  the  arc  between  the  ex- 
tended sides,  add  it  to  the  arc  of  the  opposite  or  vertical  angle, 
and  compare  the  sum  with  the  measure  of  the  angle.  How 
does  the  measure  of  the  angle  compare  with  the  sum  of  the 
two  arcs  ?   What  general  inference  may  be  drawn  from  this  ? 

9.  What  is  the  measure  of  an  interior  eccentric  angle 
whose  arcs  are  58°-17°,  69°-23°,  78°-31°,  49°-13°,  85°-29°, 
or  106°-47°? 

10.  What  is  one  of  the  arcs  of  an  interior  eccentric  angle 
when  the  other  is  59°,  and  the  angle  69°,  57°,  98°,  117°, 
123°,  or  79°  ? 

11.  Upon  the  arc  of  a  circle,  erect  an  exterior  eccentric  angle 
—  an  angle  whose  vertex  lies  without  the  circumference. 
Find  the  measure  of  this  angle.  Measure  off  on  the  arc 
upon  which  the  angle  stands  an  arc  equal  to  the  other  arc 
intercepted  by  the  sides  of  the  angle,  to  find  the  differ- 
ence between  the  arcs.  Compare  the  measure  of  the  angle 
with  this  difference.     Draw  the  inference. 

12.  How  many  degrees  in  an  exterior  eccentric  angle,  if 
the  arcs  between  its  sides  are  115°-47°,  98°-17°,  101°-59°, 
83°-19°,  99°-48°,  or  123°-36°  ? 

13.  If  one  of  the  arcs  between  the  sides  of  an  exterior 
eccentric  angle  is  29°,  what  is  the  other  if  the  angle  is  24°, 
36°,  48°,  19°,  67°,  or  78°  ? 

2.   The  Diameter  and  the  Circumference  of  a  Circle. 

1.  In  the  year  212  B.C.,  Archimedes,  the  greatest  of 
ancient  mathematicians,  found  that  the  circumference  of 
a  circle  is  very  nearly  3.14  (more  nearly  3.1416)  times  the 
diameter.  When,  therefore,  the  diameter  is  given,  how  is 
the  circumference  found  ?  If  the  circumference  is  given, 
how  is  the  diameter  found? 


the  ant  i.e.  43 

2.  Bind  the  circumference  of  a  circle  whose  diameter  is 
6,  7.  8,  12,  20,  ox32  inches. 

3.  What  is  the  diameter  of  a  circle  whose  circumference 
is  50.24,  70.48,  120.36,  212.75,  or  342.62  inches? 

3.     The  Area  op  the  OlBOLB. 

1.  By  means  of  radii,  divide  a  circle  into  triangles. 

2.  If  the  arcs  upon  which  the  radii  stand  be  considered 
as  the  bases  of  the  triangles,  what  line  of  the  circle  is  the 
sum  of  the  bases  ? 

3.  What  line  is  the  common  height  or  altitude  of  the 

triangles? 

4.  The  sum  of  the  areas  of  the  triangles,  and  therefore 
the  area  of  the  circle,  is  found  by  multiplying  the  circum- 
ference by  half  the  radius.     The  following  is  the  formula: 

Remark.  —  The  pupils  should  be  led  to  discover  the  method  of 
rinding  the  area  of  the  circle  from  the  sum  of  the  triangles. 

5.  Since  the  circumference  is  3.14  times  the  diameter, 
the  area  of  the  circle  may  also  be  found  by  taking  3.14 

the  diameter  and  multiplying  it  by  half  the  radius: 

314     £     22 

100  X  1  X  2 ' 

6.  Since,  however,  the  diameter  is  twice  the  radius,  the 

foregoing  formula  may  be  written,  — —  x X  — ,  and  by 

canceling  the  2's  in  the  numerator  and  the  denominator, 
may  be  reduced  to  gj  *  *  x  *  0r  g£  x  * 


44  CONSTRUCTIVE  GEOMETRY. 

7.  What  factor  must  be  known  that  the  area  may  be 
found  by  the  last  formula? 

8.  If  the  diameter  of  a  circle  is  4,  8,  9,  12,  15,  18,  or  20 
inches,  what  is  the  area  ? 

9.  If  the  area  of  a  circle  is  40.18,  84.96,  184.14,  456.24, 
or  378.16  square  inches,  what  is  its  diameter? 

4.     The  Annulus,  or  Circle-ring. 

1.  With  the  same  center,  describe  two  circles  of  unequal 
diameters.     Such  circles  are  called  concentric  circles. 

2.  Within  a  circle,  describe  another  circle  which  shall 
not  have  the  same  center  as  the  other.  Such  circles  are 
called  eccentric  circles. 

3.  The  surface  between  two  concentric  or  two  eccentric 
circles  is  called  an  annulus,  or  circle-ring. 

4.  If  you  had  given  the  area  of  each  of  two  concentric 
or  eccentric  circles,  how  would  you  find  the  area  of  the 

circle-ring  ? 

5.  Find  the  areas  of  the  following  circle-rings,  the  diame- 
ters of  the  two  circles  being  given  : 

Diam.  Larger.  Diam.  Smaller. 

(1)  15  cm.  8  cm. 

(2)  24  cm.  17  cm. 

(3)  16  cm.  9  cm. 

6.  The  areas  of  two  concentric  circles  being  given,  to 
find  the  width  of  the  circle-rings  : 

Larger.  Smaller.  Larger.  Smaller. 

(1)  452.16  sq.  m.       50.24  sq.  m.  (3)     803.84  sq.  m.     200.96  sq.  m. 

(2)  615.40  sq.  m.       113.04  sq.  m.         (4)  1256.00  sq.  m.    314.00  sq.  m. 

7.  Given  the  areas  of  two  eccentric  circles,  the  smaller 
being  tangent  to  the  larger,  to  find  the  width  of  the  circle- 
rings  in  the  central  lines : 


Diam.  Larger. 

Diam.  Smaller. 

(4)  28  cm. 

13  cm. 

(5)  19  cm. 

6  cm. 

(6)  26  cm. 

12  cm. 

Tin:  CIRCLE,  45 

Lai.                                   !!<t  Larjjir.  Smaller. 

I     879  M  sM.  in.      1  •_».:.«;  sq.  in.  (3)  7<>f,  r,o  s,,.  m.  i:,:;.st;  >,,.  ,„. 

9046  bq.  in.      7&60  iq.  m.  (4)  907.40  s.j.  m.  86184  Bq.  m. 


5.    Tiik  Sector, 

1.  Draw  two  radii  in  a  circle.  The  part  of  the  surface 
included  between  the  radii  and  the  intercepted  arc  is  called 

2.  Draw  a  sector  whose  arc  is  90°.  This  sector,  contain- 
in-  <»ne  fourth  of  the  area  of  the  circle,  is  called  a  quadrant 

3.  A  sector  whose  arc  is  60°  is  called  a  sextant,  and  one 
whose  arc  is  4o°,  an  octant. 

4.  Find  the  area  of  the  quadrant,  sextant,  and  octant, 
when  the  diameter  of  the  circle  is  18,  28,  48,  72,  84,  or 
L24  cm. 

5.  To  find  the  area  of  a  sector  Mien  the  number  of  degrees 
in  its  arc  is  not  an  exact  divisor  of  the  circumference.  —  Find 
tli«'  area  of  a  sector  of  one  degree  of  arc,  and  multiply  it  by 
the  number  of  degrees  in  the  given  arc. 

6.  Find  the  area  of  each  of  the  following  sectors,  the 
arcs  and  diameters  being  given  \ 

Arc.  DinmcU-r.  Arc.  Di.im.tor. 

(1)  57°  8  cm.  (4)     87°  48  cm. 

(2)  108°  18  cm.  (5)   119  27  cm. 

(3)  95°  24  cm.  (6)     46°  39  cm. 

(J.     Tiik  Sk<.mi:\  r. 

1.  Draw  a  chord  in  a  circle.  The  part  of  the  area  of  the 
circle  that  lies  between  the  chord  and  its  arc  is  called  a 
segment. 

2.  If  you  had  given  the  arc,  the  Chord,  and  the  diameter 
of  the  circle,  could  you  find  the  area  of  the  segment  ?     Could 


46  CONSTRUCTIVE  GEOMETRY. 

you  find  the  area  of  the  isosceles  triangle  whose  base  is  the 
chord  ? 

3.  Find  the  area  of  each  of  the  following  segments,  the 
radius  of  the  circle,  the  arc,  the  chord,  and  the  altitude  of 
the  triangle  being  given  : 


Radius. 

Arc. 

Chord. 

Alt. 

(1) 

7.2  cm. 

63° 

7.5  cm. 

6.1  cm. 

(2) 

5.6  cm. 

106° 

9.1  cm. 

5.3  cm. 

(3) 

5.9  cm. 

58° 

5.8  cm. 

5.2  cm, 

(4) 

10.1  cm. 

53° 

9.1  cm. 

9  cm. 

(5) 

5.3  cm. 

147° 

10  cm. 

1.9  cm. 

(6) 

9.3  cm. 

55° 

8.7  cm. 

8.3  cm. 

IX.    THE    FUNDAMENTAL    MATHEMATICAL 
BODIES. 

Introduction. 

1.  The  floor,  ceiling,  and  four  sides  of  this  room  inclose 
a  certain  amount  of  space  or  room. 

2.  Any  portion  of  space  inclosed  on  all  sides  (including 
floor  and  ceiling)  is  called  a  geometrical  body. 

3.  What  kind  of  geometrical  figures  are  the  sides  of  this 
room  ?     What  kind  of  parallelograms  are  they  ? 

4.  What  direction  do  the  ceiling  and  the  floor  take  ? 
What  direction  with  reference  to  each  other  ? 

5.  What  direction  do  the  sides  and  ends  take?  What, 
with  reference  to  each  other  ? 

6.  Compare  the  size  of  the  floor  with  that  of  the  ceiling. 
Compare  also  the  sides  and  the  ends  with  each  other  as  to 
size. 

7.  What  may  be  said  of  the  size  of  any  two  parallel 
inclosures  or  sides  of  a  room  ? 


MATHEMATICAL    BODIES.  47 

8.  In  what  kind  of  form  do  the  floor  and  the  sides  inter- 
sect each  other?     In  what,  the  ceiling  and  the  sides? 

9.  The  line  of  intersection  in  which  two  sides  meet 
each  other  is  called  an  edge  ;  it  is  also  called  the  axis  of 
tli,-  btieneeUng  sides. 

10.  What  kind  of  angle  do  any  two  intersecting  sides  of 
this  room  make  with  each  other?  How  many  such  angles 
can  you  find  in  the  room? 

11.  Since  the  interior  angle  formed  by  two  sides  is  a 
right  angle,  what  kind  of  angle  is  the  exterior  angle  formed 
l»v  two  adjacent  sides  of  a  house?  How  many  right  angles 
does  it  contain  ? 

12.  How  many  corners  has  this  room  ?  How  many  sur- 
faces or  sides  (including  floor  and  ceiling)  form  the  corners  ? 

13.  Since  the  angles  of  the  parallelograms  that  form  the 
corner  angles  are  right  angles,  what  kind  of  angles  may 
the  corner  angles  be  called  ? 

14.  A  line  connecting  the  middle  points  of  two  parallel 
sides  is  a  surface  axis.  How  many  such  are  possible  in  this 
room? 

15.  A  line  connecting  the  middle  points  of  two  opposite 
edges  or  axes  is  a  line  axis.  How  many  such  are  possible 
in  this  room  ? 

16.  A  line  connecting  two  opposite  corners  is  a  corner 
axis.     How  many  such  are  possible  in  this  room  ? 

17.  Find  the  area  of  the  six  sides  of  this  room.  Their 
sum  is  the  whole  surface  of  the  room. 

18.  Find  the  surface  area  of  a  room  whose  dimensions 
are:  length  6.6*  m.,  width  4  m.,  height  3.5  m. 


48  CONSTRUCTIVE  GEOMETRY. 

THE   SURFACES   OF   GEOMETRICAL  BODIES. 
1.     The  Regular  Bodies. 
The  Cube.  —  1.    How  many  sides  has  the  cube  ? 

2.  To  what  class  of  figures  do  the  sides  of  the  cube 
belong  ? 

3.  How  do  the  sides  compare  in  size  ? 

4.  How  could  you  find  the  surface  of  a  cube,  if  only  its 
side  or  edge  were  given? 

5.  If  the  surface  of  a  cube  were  given,  how  would  you 
find  the  length  of  a  side  ? 

6.  What  is  the  surface  of  a  cube  whose  edge  is  6,  9,  12, 
16,  20,  or  24  cm.  ? 

7.  If  the  surface  of  a  cube  is  294,  864,  1014,  486,  1536, 
or  384  sq.  cm.,  what  is  its  edge  ? 

The  Tetrahedron.  —  The  tetrahedron  is  a  solid  inclosed 
or  bounded  by  four  equal  equilateral  triangles. 

Note. — An  equilateral  triangle  is  one  having  all  its  sides  equal. 

The  Octahedron.  —  The  octahedron  is  a  solid  bounded  by 
eight  equilateral  triangles. 

The  Icosahedron.  —  The  icosahedron  is  a  solid  bounded 
by  twenty  equal  equilateral  triangles.  It  is  a  solid  com- 
posed of  twenty  equal  and  similar  triangular  pyramids 
whose  vertices  meet  in  a  common  point. 

1.  What  factors  must  be  given  to  find  the  area  of  an 
equilateral  triangle  ? 

2.  If  you  had  a  side  and  the  altitude  of  one  of  the  tri- 
angles given,  how  would  you  find  the  surface  of  any  one  of 
the  foregoing  bodies  ? 


MATHEMATICAL    B0DIM8.  49 

3.  If  the  surface  of  one  of  the  bodies  and  the  base  of 
ope  of  the  triangles  wen  given,  how  would  you  find  the 
altitude  of  the  triangle? 

4.  If  the  Burface  of  one  of  the  bodies  and  the  altitude 
<»f  a  triangle  were  given,  how  would  you  find  the  be 

the  triangle? 

5.  Find  the  Burface  of  each  <>ne  of  the  foregoing  bodies. 
the  base  and  altitude  of  one  of  its  triangles  being  gives  : 

Altitll.tr. 

(1)  lent  3.46  em. 

(2)  C».4  cm.  5.54  cm. 

(3)  8.4  cm.  7.27  cm. 

6.  What  is  the  altitude  of  one  of  the  triangles  of  an 
octahedron,  if  the  surface  of  the  body  is  280.584  Bq.  cm;, 

and  the  base  of  one  of  its  triangles  1)  cm.? 

7.  Find  the  base  of  a  triangle  of  an  Leosahedron  whose 

surface  is  2216.96  sq.  Cm.,  and  the  altitude  of  one  of  its 
triangles  13.856  em. 

The  Dodecahedron.  — The  dodecahedron  is  a  solid  bounded 

by  twelve  equal  regular  pentagons  (live-sided  figures). 

1.  How  can  you  find  the  area  of  a  pentagon  ? 

2.  If  you  should  inscribe  a  circle  in  a  pentagon,  and 
draw  straight  lines  from  its  center  to  the  angles  or  corners 

of  the  pentagon,  into  what  kind  of  figures  would  you  divide 
the  pentagon? 

3.  If  you  had  given  the  side  of  one  of  the  pentagons  and 
the  radius  of  the  inscribed  circle,  how  would  you  find  the 
surface  of  the  dodecahedron  '.' 

4.  If  you  had   given   the  surface  of  a  dodecahedron  and 

tin*  side  of  one  of  its   pentagons,  how   would  you   find    the 
radius  of  the  inscribed  circle  '.' 


50  CONSTRUCTIVE  GEOMETRY. 

5.  Find  the  surface  of  a  dodecahedron,  the  side  of  one 
of  whose  pentagons  is  8  cm.,  and  the  radius  of  the  inscribed 
circle  5.5  cm. 

6.  The  surface  of  a  dodecahedron  is  743.4  sq.  cm.,  the 
side  of  one  of  its  pentagons  6  cm. ;  what  is  the  radius  of 
the  inscribed  circle  ? 

7.  The  surface  of  a  dodecahedron  is  1671.3  sq.  cm.,  the 
radius  of  the  inscribed  circle  of  a  pentagon  6.19  cm. ;  what 
is  the  side  of  a  pentagon  ? 

The  Sphere,  or  Ball.  —  Not  only  are  polygons,  whose  sides 
and  angles  are  equal,  regarded  as  regular  figures,  but 
likewise  squares,  equilateral  triangles,  and  all  solids  whose 
surfaces  are  composed  of  equal,  regular  figures.  The  cube, 
tetrahedron,  octahedron,  icosahedron,  and  dodecahedron  are 
regarded  as  regular  solids. 

1.  The  sphere  may  also  be  considered  a  regular  body  or 
solid.  It  is  inclosed  in  a  regularly  curved  surface,  every 
point  of  which  is  equally  distant  from  a  point  within  called 
the  center. 

2.  Lines  drawn  from  the  center  to  the  surface  are  radii 
of  the  sphere.     How  do  these  compare  in  length  ?     Why  ? 

3.  A  straight  line  passing  through  the  center,  and  limited 
at  both  extremities  by  the  surface,  is  a  diameter  or  sphere 
axis,  and  its  terminal  points  or  ends  are  called  poles.  How 
does  the  diameter  compare  in  length  with  the  radius? 
What  inference  may  be  drawn  from  this? 

4.  If  you  should  take  a  straight,  sharp  knife  and  cut  a 
sphere  through  its  center,  into  what  kind  of  parts  would 
you  divide  it  ? 

5.  What  kind  of  figure  would  the  cut  surface  make  ? 


MATVXMATICAL   B0DIB8.  51 

6.  In  what  respects  would  it  agree,  or  correspond,  with 
tlu'  sphere  ? 

7.  Four  time*  the  area  of  the  owl  turfoG  (great  circle)  is 
tin'  eurfaee  of  the  tphere, 

8.  It'  the  diameter  alone  of  a  sphere  were  given,  how 
would  you  find  tin*  surface? 

9.  On  page  43  we  have  a  formula  for  finding  the  area  of 
:i  circle.  If  in  that  formula  we  substitute  the  square  of 
half  the  diameter  for  the  square  of  the  radius,  we  have  for 
the  surface  of  the  sphere  the  following  formula: 

100  X  2  X  2  X  1'  °    100  X  1  ' 

10.  If  the  surface  of  a  sphere  were  given,  how  would  you 

find  the  diameter? 

11.  Find  the  surface  of  a  sphere  whose  diameter  is  6,  8, 
16,  is.  20,  or  L'l  em. 

12.  The  diameter  of  a  ball  is  3  inches;  what  is  its  surface  ? 

13.  If  the  surface  of  a  sphere  is  96.5843,  132.G894,  or 
96  1.9876  sq.  em.,  what  is  its  diameter? 

2.   The  Half-regular,  Uniformly-thick,  Bodies. 

The  Pillar  or  Prism.  — 1.  Examine  a  prism,  and  observe 
how  many  regular  figures  its  surface  has. 

2.  Of  what  kind  are  they  in  a  three,  four,  five,  or  six- 
sided  prism  ? 

3.  How  do  the  parts  of  each  compare  in  size  ? 

4.  How  can  we  find  the  sum  of  the  surfaces  of  all  the 
figures  of  each  ? 

5.  By  how  many  surfaces  is  eaeh  of  the  prisma  bounded  f 


52  CONSTRUCTIVE  GEOMETRY. 

6.  Of  which  of  them  are  the  surfaces  inclined  toward 
each  other  ? 

7.  What  kind  of  figures  are  the  inclined  surfaces  ?     How 
can  you  tell  ? 

8.  How  do  these  surfaces  compare  with  each  other  in 
form  and  size  ? 

9.  What  factor  have  the  sides  and  the  bases  of  the  prisms 
in  common  ? 

10.  What  is  the  entire  surface  of  a  triangular  prism,  the 
side  of  whose  base  (side  of  triangle)  is  5  cm.,  the  perpen- 
dicular upon  it  from  the  vertex  of  the  opposite  angle  4.4  cm., 
and  the  height  of  the  prism  18  cm.  ? 

11.  What  is  the  entire  surface  of  a  triangular  prism,  when 
the  side  of  the  base  is  12  cm.,  the  perpendicular  from  the 
vertex  of  the  opposite  angle  10.4  cm.,  and  the  height  of  the 
prism  3.6  cm.  ? 

12.  What  is  the  surface  of  a  square  prism  or  pillar,  the 
side  of  whose  base  is  12  cm.  and  height  80  cm.  ? 

13.  What  is  the  surface  of  a  regular  five-sided  (pentago- 
nal) column  whose  height  is  80  cm.,  side  of  base  9.6  cm., 
and  the  perpendicular  upon  one  of  the  sides  from  center 
6.6  cm.  ? 

14.  What  is  the  entire  surface  of  a  regular  hexagonal 
(six-sided)  column  whose  height  is  56  cm.,  length  of  a  side 
of  the  base  6  cm.,  and  the  radius  of  the  inscribed  circle, 
5.196  cm.  ? 

Kemark.  —  Since  prisms  are  inclosed  by  two  equal  regular  figures 
and  by  equal  rectangles,  they  are  called  half-regular  bodies  ;  and  since 
their  perimeters  (measures  around)  are  everywhere  the  same,  they 
are  called  uniformly  thick  bodies. 


MATHEMATICAL   BODIES.  53 

The  Round  Pillar  or  Cylinder. —  1.  How  many  equal  faces 
Ot  surfaces  has  the  cylinder '.' 

2.  What  kind  of  figures  are  they  ? 

3.  How  would  you  find  the  sum  of  their  surfaces  ? 

4.  What  kind  of  figure  would  the  surface  between  the 
ends  form  if  it  were  unrolled  ?  How  would  you  find  its 
area  ? 

5.  What  factor  have  the  ends  and  the  side  surface  in 
common  ? 

Hi  m  \i;k.  —  When  the  cylinder  is  lying  horizontally  it  is  called  a 
roller,  md  when  it  stands  vertically,  a  round  pillar. 

6.  The  diameter  of  a  cylinder  is  12  cm.  and  its  height 
80  cm. ;  what  is  its  surface  ? 

7.  A  roller  is  120  cm.  in  length  and  18  cm.  in  thickness; 
what  is  its  surface  ? 

8.  What  is  the  surface  of  a  roller  that  is  12  cm.  in  thick- 
ness and  84  cm.  in  length  ? 

9  What  is  the  surface  6f  a  cylinder  whose  diameter  is 
18  cm.  and  height  1.6  cm  ? 

.*!     Tin.    H  mi  i;i  (jular,   Tapering   or  Pointed    Bodies. 

The  Pyramid. — 1.  How  many  regular  figures  has  a 
pyramid  ? 

2.  What  kind  of  regular  figures  are  found  in  a  three, 
four,  five,  or  six  sided  pyramid  ? 

3.  How  would  you  find  their  area,  or  surface  ? 

4.  How  many  separate  surfaces  has  each  of  the  foregoing 
pyramids  ? 

5.  Of  which  of  them  are  the  sides  inclined  towards  each 

other? 


54  CONSTRUCTIVE  GEOMETRY. 

6.  What  kind  of  triangles  are  the  sides  of  the  pyramid  ? 

7.  How  can  the  sum  of  their  surfaces  be  found  ? 

8.  What  factor  is  common  to  the  base  and  the  sides  ? 

9.  Find  the  surfaces  of  the  following  triangular  pyra- 
mids,—  a  side  of  the  base,  the  altitude  or  perpendicular  of 
basal  triangle,  and  the  slant  height  being  given  : 


Side. 

Altitude. 

Slant  Height. 

(1) 

8  cm. 

6.93  cm. 

18  cm. 

(2) 

39  cm. 

33.775  cm. 

84  cm. 

(3) 

7.4  cm. 

6.4  cm. 

24.8  cm. 

10.  Find  the  surfaces  of  the  following  square  pyramids, 
—  a  side  of  the  base  and  the  slant  height  being  given  : 

Side.  Slant.  Height. 

(1)  6  cm.  25  cm. 

(2)  12.8  cm.  34.5  cm. 

(3)  5.8  cm.  15.4  cm. 

11.  Find  the  surfaces  of  the  following  regular  pentagonal 
(five-sided)  pyramids,  —  a  side  of  the  base,  the  radius  of  the 
inscribed  circle,  and  the  slant  height  being  given  : 

Side  of  Base.  Radius.  Slant  Height. 

(1)  12.8  cm.  8.8  cm.  72  cm. 

(2)  25.4  cm.  17.48  cm.  92  cm. 

12.  Find  the  surfaces- of  the  following  regular  hexagonal 
(six-sided)  pyramids,  —  a  side  of  the  base,  the  radius  of  the 
inscribed  circle,  and  the  slant  height  being  given: 

Side  of  Base.  Radius.  Slant  Height. 

(1)  2.4  cm.  2.0784  cm.  56  cm. 

(2)  7  cm.  6.06  cm.  36  cm. 

Remark.  —  The  base  alone  of  the  pyramid  is  a  regular  figure,  the 
sides  being  isosceles  triangles.  The  pyramid  is  also  a  half-regular 
body  ;  but,  since  its  triangular  sides  taper  to  a  point,  it  is  also  called  a 
half-regular  pointed  body. 

A  perpendicular  from  the  apex  to  the  base  is  the  altitude. 
and  is  called  the  axis  of  the  pyramid. 


M.  I  Tin: MA  TH  \ 1 1    BODIES.  55 

The  Cone.  —  1.    How  many  Burfaoea  lias  the  cone  ? 

2.  What  kind  of  figure  is  the  base? 

3.  How  is  tin-  area  of  the  base  found  '.' 

4.  What  kind  of  figure  is  the  convex  surface  ? 

5.  What  kind  of  figure  would  it  resemble  if  it  were 
unrolled  ? 

6.  How  is  the  area  of  the  convex  surface  found  ? 

7.  To  what  line  of  the  base  does  the  curve  of  the  convex 
surface  correspond '.' 

Nmi:.  —  A  line  from  the  vertex  of  the  cone  to  the  middle  of  the 
/  thr  cone,  and,  in  a  perpendicular  cone,  is  also  the 
altitude. 

Tin  Hani  height  of  a  cone  is  the  distance  from  the  apex  to  the  <n- 
enmferenoe  of  the  base. 

8.  What  is  the  surface  of  each  of  the  following  con 

the  diameter  of  the  base  and  the  altitude  (height)  being 
given '.' 

I>i:iin.  of  Rase.  Altitude.  I  Mum.  of  Rase.  Altitude. 

(1)  4  cm.  9  cm.  (4)     8.6  cm.        36  cm. 

(2)  6  cm.         19.4  mi.  (5)       12  cm.         52  cm. 

(3)  7.5  cm.        10.6  cm.  (6)    15.6  cm.        62  cm. 

9.  What  is  the  surface  of  a  cone,  the  circumference  ol 
wht.se  base  is  15.7  cm.  and  altitude  48  cm.  ? 


4.  Tin-  Truncated,  ob  Shortened,  Bodies. 

Note.  —  If  from  a  pointed  body  (pyramid  or  cone)  apiece  bo  cut 
off  parallel  t<>  the  baee,  the  remainder  d  ■  thortened  hody. 

The  only  shortened  bodies  that  will  be  considered  are  the  square 
pyramid  and  the  cone. 


56  CONSTRUCTIVE  GEOMETRY. 

The  Shortened  Square  Pyramid.  —  1.  By  how  many  sur- 
faces is  the  shortened  square  pyramid  bounded  ? 

2.  What  kind  of  figures  are  the  ends  or  bases  ? 

3.  In  what  respects  do  they  differ  ? 

4.  What  factors  are  necessary  to  determine  their  sur- 
faces ? 

5.  Of  what  kind  of  figures  are  the  sides  composed  ? 

6.  How  do  the  sides  compare  in  size  ? 

7.  What  factor  is  yet  necessary  to  determine  their  sur- 
faces ? 

8.  Find  the  surfaces  of  the  following  shortened  square 
pyramids,  —  a  side  of  the  lower  base,  a  side  of  the  upper 
base  or  end,  and  the  slant  height  being  given  : 


Side  of  Lower  Base. 

Side  of  Upper  Base. 

Slant  Height. 

(1)      3  cm. 

2  cm. 

12  cm. 

(2)      5  cm. 

4  cm. 

16  cm. 

(3)  5.8  cm. 

4.2  cm. 

12.5  cm. 

The  Shortened  Cone.  —  1.    By  how  many  surfaces  is  the 
shortened  cone  bounded  ? 

2.  What  kind  of  figures  form  the  top  and  the  bottom  ? 

3.  In  what  respects  do  the  top  and  the  bottom  differ  ? 

4.  How  is  the  surface  of  each  found  ? 

5.  What  kind  of  figure  would  the  convex  envelope  form 
if  it  were  unrolled  upon  a  plane  surface  ?     A  trapezoid. 

6.  What  kind  of  lines  would  the  parallel  sides  form  ? 
Arcs  of  circles. 

7.  To  what  lines  are  these  arcs  equal  ? 

8.  How,  by  means  of  these  arcs,  may  the  convex  surface 
be  found  ? 


MATHEMATICAL   BODI1  57 

9.  Kind  the  surfaces  of  the  following  shortened  cones, — 
tlic  diameter  oi  the  lower  base,  of  the  upper  base,  and  the 
slant,  height  being  given  : 

rOtam.  Upper  Dfaun.  slant  II.                      m  Diam.  Upper  Dim.  Slant  II. 

(1)      9  cm.           5  cm.      24  cm.  (4)  3.8  cm.  3  cm.     6.6  cm. 

I     1.4  cm.          .8  cm.       1(5  cm.  (6)  4.8  cm.  4.6  cm.      4.2  cm. 

(3)  2.4  cm.         .3  cm.      3.8  cm.  (6)  4.4  cm.  4  cm.    I.Mem. 


THE   CONTENTS   OF   SOLIDS,    <>K    BODIES. 
1.    Bodies  of  Unifokm  Thioknbs«. 

1.  What  is  the  length  of  a  side  of  a  cubic  inch?  Of 
a  cubic  foot?  A  cubic  centimeter?  A  cubic  decimeter? 
A  cubic  meter? 

2.  How  many  cubic  centimeters  can  stand  by  the  side 
of  <>ne  another  on  the  base  of  any  body  or  solid  ? 

3.  How  high  is  one  layer  of  cubic  centimeters? 

4.  How  many  such  layers  are  necessary  to  fill  any 
portion  of  space? 

5.  I  low  would  you  find  the  cubic  contents  of  any  por- 
tion of  space? 

6.  Find  the  contents  of  a  cube  whose  side  is  8,  9,  10,  12, 
1  I.  or  L6  em. 

7.  Find  the  cubic  contents  of  the  following  bodies: 

(a)  A  regular  triangular  prism,  a  side  of  whose  base  is 
8  <  m.,  perpendicular  of  basal  triangle  from  vertex  of  an 
angle  to  opposite  side  6.9  em.,  and  height  of  prism  24  cm. 

(b)  A  regular  pentagonal  column  whose  heighl  is  134  cm., 
a  side  of  base  8  cm.,  and  radius  of  inscribed  circle  5.6  cm. 

(c)  A  regular  hexagonal  column  whose  height  is  lL'n 
cm.,  a  side  of  base  8.4  em.,  and  radius  of  inscribed  circle 
7.3  cm. 

(d)  A  roller  whose  length  is  6.4  cm.  and  diameter  13  cm. 


58  CONSTRUCTIVE  GEOMETRY. 

8.  The  cubic  contents  of  a  roller  9  dm.  in  length  are 
11 3.01  cu.  dm.,  what  is  its  diameter  ? 

9.  What  are  the  contents  of  the  solid  part  of  a  hollow 
cylinder  whose  diameter  is  6.4  cm.,  diameter  of  hollow  .part 
5  cm.,  and  height  120  cm.  ? 

2.   Tapering  or  Pointed  Bodies. 

1.  The  cubic  contents  of  a  pointed  or  tapering  body  are 
one  third  of  those  of  a  regular  solid  of  the  same  base  and 
altitude  as  the  tapering  body. 

2.  Find  the  cubic  contents  of  the  following  bodies  : 

(a)  A  regular  triangular  pyramid,  —  a  side  of  the  base, 
perpendicular  upon  side  from  vertex  of  opposite  angle, 
and  altitude  of  pyramid  being  given  : 

Side  of  Base.  Perpendicular.  Altitude. 

(1)  4.6  cm.  3.98  cm.  8  cm. 

(2)  92.3  cm.  79.9    cm.  3G0  cm. 

(6)  A  square  pyramid, — a  side  of  its  base  and  the  altitude 
being  given : 

Side  of  Base.  Altitude. 

(1)  7  cm.  17  cm. 

(2)  82  cm.  240  cm. 

(c)  A  regular  pentagonal  pyramid  whose  side  of  base, 
radius  of  inscribed  circle,  and  altitude  are  given. 

Side  of  Base. 

(1)  4.8  cm. 

(2)  34.5  cm. 

(d)  A  regular  octagonal  (eight-sided)  pyramid  whose 
altitude  is  35  cm.,  a  side  of  base  6.4  cm.,  and  radius  of 
inscribed  circle  7.73  cm. 

(e)  A  cone  whose  diameter  and  altitude  are  given. 


Eadius. 

Altitude. 

3.3  cm. 

36  cm. 

23.7  cm. 

180  cm. 

Diameter. 

Altitude. 

(1)     23  cm. 

69  cm. 

(2)    3.8  dm. 

74  dm. 

MATHEMATICAL    BODIES,  69 

3.  Cukic  Contents  oi  Tbi  m  ltkd,  ob  Bhobtbited, 
Bodies, 

1.  The  cubic  contents  of  a  shortened  cone  may  be  found 
with  Buffioient  exactness  for  all  practical  purposes  (approxi- 
mately), when  the  difference  between  the  upper  and  tin* 
lower  surface  is  small,  by  finding  the  middle  diameter  be- 
tween the  two  ends  (half  their  sunn,  and  treating  the  body 
M  a  cylinder  whose  diameter  is  the  middle  diameter  and 
height  that  of  the  shortened  cone. 

2.  If  the  lower  diameter  of  a  shortened  cone  is  4  cm.,  the 
upper  .')  cm.,  and  the  height  6  cm.;  what  are  its  contents'.' 

3.  The   result  by  the  foregoing   method  is  a   little    too 

small.  When  exactness  is  required,  the  following  method 
may  be  pursued:  From  the  whole  cone  subtract  the  part 
cut  off.  But  to  subtract  the  part  cut  off,  the  contents  ami 
altitude  of  the  whole  cone  must  be  found.  The  altitude 
may  lie  found  by  the  following  proportion:  As  the  differ- 
ence of  the  radii  is  to  the  larger  radius  (radius  of  cone),  so 
is  the  height  of  the  shortened  part  to  the  height  of  the 
whole  cone. 

The  height  of  the  whole  cone  may  be  directly  found  by 
the  rule  derived  from  the  foregoing  proportion,  namely: 
Multiply  the  larger  radius  by  the  height  of  the  shortened 
body,  and  divide  the  product  by  the  difference  between  the 
two  radii. 

4.  Taking  the  foregoing  problem,  we  have  2x6xf  =  24, 
the  height  of  the  whole  cone.  Subtracting  from  this  the 
height  of  the  shortened  part,  6,  we  have  18,  the  height 
of  the  part  cut  off.  To  find  the  contents  of  the  whole  cone, 
we  have  2  x  2  x  3.14  x  ^.  To  find  the  contents  of  the 
part  cut  off.  we  have  f  x}x  3.14  x  J/»  an<*  tne  difference 
is  the  shortened  part. 


60  CONSTRUCTIVE  GEOMETRY. 

5.  Find  the  contents  of  the  following  shortened  cones 
by  both  methods: 

Larger  Diam.   Shorter  Diam.   Height.  Larger  Diam.   Shorter  Diam.    Height. 

(1)  8  cm.     6  cm.    9  cm.    (4)  24  cm.    22  cm.   180  cm. 

(2)  6  cm.     4  cm.    9  cm.    (5)  48  cm.    44  cm.   120  cm. 

(3)  28  cm.    2G  cm.   98  cm.    (0)  38  cm.    30  cm.    58  cm. 

6.  The  contents  of  a  barrel  may  also  be  found  by  the 
foregoing  method;  the  larger  diameter  being  the  one  at 
the  bung  (half-way  between  the  ends),  the  shorter  that  at 
the  ends,  and  the  height  the  half  length  of  the  barrel. 

7.  Find  the  contents  of  the  following  barrels  by  both  of 
the  foregoing  methods : 


Bung  Diam. 

End  Diam. 

Length. 

(1)  38  cm. 

34  cm. 

40  cm. 

(2)  48  cm. 

45  cm. 

58  cm. 

(3)  38  cm. 

35  cm. 

42  cm. 

8.  The  contents  of  a  shortened  square  pyramid  may,  for 
all  practical  purposes,  be  found  by  multiplying  the  half 
sum  of  the  areas  of  the  ends  by  the  height. 

9.  What  are  the  contents  of  a  shortened  square  pyramid 
whose  height  is  8  dm.,  side  of  base  3.8  dm.,  and  of  top 
3.4  dm.  ? 

10.  When  exactness  is  required,  the  following  formula 
should  be  used,  in  which  A  stands  for  the  area  of  the  larger 
end,  a  for  that  of  the  smaller,  S  for  a  side  of  the  larger,  s 
for  that  of  the  smaller,  and  H  for  the  height : 

Remark.  —In  this  formula,  the  expression  (S  x  s)  stands  for  the 
square  root  of  the  product  of  the  areas  of  the  ends. 

11.  With  this  formula,  find  the  contents  of  the  following 
shortened  square  pyramids : 


Area  of  Larger  End. 

Area  of  Smaller  End. 

Height. 

(1)  12  cm. 

10  cm. 

45  cm. 

(2)  10  cm. 

14  cm. 

58  cm. 

MATHEMATICAL   BODU  61 

12.    If  the  end  surfaces  of  a  Bbortened   pyramid  are  not 
squares,   bui  three,  five,  or   more  sided  figures,  to  obtain 
bnesa  the  following  formula  must  be  used: 


// 


\A  +  a  +  V(A  x  a)] 


Reg U LAB    I!<»i»ii:>. 


1.  The  tetrahedron  (four-surfaced  body)  belongs  to  the 
tapering  or  pointed  bodies.  Its  contents  are  therefore 
found  by  multiplying  its  base  by  a  third  of  its  height. 

2.  Find  the  contents  of  the  following  tetrahedrons  : 


Si.lc.f  Base. 

Perpend.  <>t  K:i-:il  Triangle. 

Altittl.lr. 

(1)  4  cm. 

8.46  cm. 

3.26  cm. 

(2)  5  cm. 

:  cm. 

4.08  cm. 

(8)  5  cm. 

5.2    cm. 

4.9    cm. 

3.  The  octahedron  (eight-surfaced  body)  can  be  divided 
into  eight  equal  pyramids,  all  of  whose  vertices  meei  at  the 
middle  point  of  the  body,  and  all  of  the  pyramids  haying 
equal  bases  and  equal  altitudes.  The  contents  of  any  <>i 
the  pyramids  are  found  by  multiplying  the  base  by  a  third 
<>f  the  altitude,  and  the  contents  (of  the  whole  body)  of  all 
the  pyramids,  by  multiplying  the  contents  of  one  of  them 
by  8,  the  number  of  them.  The  contents  of  the  octahedron 
may  also  be  found  by  multiplying  the  sum  of  the  bases  of 
the  pyramids  (the  entire  surface  of  the  body)  by  a  third  of 
the  height  of  one  of  them.  Bui  since  the  height  of  a  pyra- 
mid is  the  same  as  a  half  of  its  surface  axis,  the  contents  of 
the  octahedron  may  be  found  by  multiplying  its  surface  by 
a  sixth  of  its  surface  axis.     For  the  same  reason,  the  contents 

of  an  icosahedron  (twenty  plane-faced)  and  of  a  dodeca- 
hedron (twelve  plane-faced)  may  be  found  by  multiplying 
the  surface  by  a  sixth  of  the  surface  axis. 


62  CONSTRUCTIVE  GEOMETRY. 

5.    The  Ball,  or  Sphere. 

4.  The  last  of  the  regular  bodies  is  the  ball,  or  sphere. 
This  body  may  be  considered  as  composed  of  an  infinite 
number  of  pyramids,  the  sum  of  whose  bases  constitutes  the 
surface,  and  the  height  or  altitude  of  the  pyramids  the 
radius  of  the  ball.  The  contents  of  a  ball  are  therefore 
equal  to  the  product  of  its  surface  by  a  third  of  its  radius 
or  a  sixth  of  its  diameter. 

5.  Since  the  surface  of  a  ball  is  found  by  the  following 

formula:   — — -  X  —  X  — ,  the  cubic  contents  are  found  by  the 
100      11  J 

following : 

314      D     D  YD_SU     D     D     #_314      D* 

100  X  1  X  1  X  6  "600  X  1  X  1  X  1  "600  X  1  ' 

6.  Find  the  contents  of  a  ball  whose  diameter  is  6,  9,  12, 
18,  30,  42,  or  56  cm. 

7.  If  the  earth  were  a  perfect  sphere,  and  its  diameter 
8000  miles,  what  would  be  its  cubic  contents  ? 

8.  Find  the  cubic  contents  of  the  following  shells  of 
hollow  balls  whose  inner  and  outer  diameters  are  given : 


Outer  Diameter. 

Inner  Diameter. 

(1)  10  cm. 

8  cm. 

(2)     5  cm. 

3.5  cm. 

(3)     9  cm. 

7.5  cm. 

SUPERIOR 

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